8 Applications to specific high-dimensional spaces. 75 .... 5. The characterization of the local structure of terminals in a Steiner minimal ..... Although Theorems 7.2 and 7.3 solve the local Steiner problem in a certain gen- ... In the Euclidean plane an angle is obsorbing if, and only if, its angular ...... with all faces congruent.
The local Steiner problem in Minkowski spaces
Konrad J. Swanepoel
Habilitationsschrift
Chemnitz, den 15. Juni 2009
c 2009 Konrad J. Swanepoel
To Annie
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Contents
Preface
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Theses
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1 A quick introduction and overview 1.1 The Steiner problem in normed spaces . . . . . . . . . . . . . . . . . . 1.1.1 The Euclidean Steiner problem . . . . . . . . . . . . . . . . . . 1.1.2 Other norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 The one-dimensional Plateau problem . . . . . . . . . . . . . . 1.2 Looking at Steiner trees locally . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Two guiding conjectures . . . . . . . . . . . . . . . . . . . . . . 1.3 Results and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 A general solution . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The two-dimensional case . . . . . . . . . . . . . . . . . . . . . 1.3.3 Cieslik’s conjecture and combinatorics . . . . . . . . . . . . . . 1.3.4 Quantitative illumination and covering . . . . . . . . . . . . . 1.3.5 A conjecture of Conger . . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Differentiable spaces (including `dp ) and Banach space theory 1.4 Restricting the number of Steiner points . . . . . . . . . . . . . . . . .
1 1 1 3 4 4 5 7 7 8 10 10 11 12 13
2 Background knowledge, notation and definitions 2.1 Signed sets . . . . . . . . . . . . . . . . . . . . 2.2 Finite-dimensional vector spaces . . . . . . . 2.3 Convexity, topology, and measure . . . . . . 2.4 Minkowski spaces . . . . . . . . . . . . . . . . 2.5 Zonotopes as unit balls . . . . . . . . . . . . . 2.5.1 General combinatorial description . . 2.5.2 Cieslik’s norm . . . . . . . . . . . . . . 2.5.3 Two-dimensional zonotopes . . . . . 2.6 Hanner polytopes and CL spaces . . . . . . . 2.7 Subdifferential calculus . . . . . . . . . . . . .
15 15 15 16 17 21 21 22 23 24 24
3 Fermat-Torricelli points and special angles
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Contents 3.1 3.2 3.3
4 The 4.1 4.2 4.3 4.4
Fermat-Torricelli points . . . . . . . . . . . . . . . . . . . . A sufficient condition for an absorbing FT configuration Critical and absorbing angles . . . . . . . . . . . . . . . . 3.3.1 General characterizations . . . . . . . . . . . . . . 3.3.2 The λ-geometry . . . . . . . . . . . . . . . . . . . . local Steiner problem Steiner minimal trees . . . . . . . . . . . . Formulation of the local Steiner problem . Degree quantities . . . . . . . . . . . . . . Other known results . . . . . . . . . . . .
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5 Solving the local Steiner problem in two dimensions 5.1 Characterization of planar terminal and Steiner configurations 5.2 Proof of the characterization . . . . . . . . . . . . . . . . . . . . 5.3 Another approach for terminal configurations . . . . . . . . . 5.4 Antipodality and higher dimensions . . . . . . . . . . . . . . .
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6 Applications to specific Minkowski planes 6.1 The Euclidean plane . . . . . . . . . . . . . . . . . . . . 6.2 Strictly convex planes . . . . . . . . . . . . . . . . . . . . 6.3 Differentiable planes . . . . . . . . . . . . . . . . . . . . 6.4 Piecewise differentiable, elliptic X-planes . . . . . . . . 6.5 An almost smooth X-plane . . . . . . . . . . . . . . . . . 6.6 A strange example . . . . . . . . . . . . . . . . . . . . . 6.7 The λ-geometry . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Degree three Steiner configurations . . . . . . . 6.7.2 Degree three terminal configurations . . . . . . 6.7.3 Degree four Steiner and terminal configurations 6.7.4 Summary . . . . . . . . . . . . . . . . . . . . . .
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7 An abstract solution in higher dimensions 7.1 Reduced Minkowski addition . . . . . . . . . . . . . . . . . . . . . . . 7.2 The characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8 Applications to specific high-dimensional spaces 8.1 The maximum degree of a terminal in MdZ . . . . . . . . . . . . . . . . 8.2 A geometric formulation of Sperner’s theorem . . . . . . . . . . . . . 8.3 Perturbations of `1d . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75 75 78 79
9 Using illumination and covering of convex bodies 9.1 Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Quantitative illumination and Steiner minimal trees . . . . . . . . . . 9.3 Quantitative covering . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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10 k-Steiner minimal trees 10.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents 10.2 1-Steiner minimal trees in Minkowski planes 10.3 Hadwiger configurations in the plane . . . . 10.3.1 Types of Hadwiger configurations . . 10.3.2 Symmetrization and Hadwiger sets . 10.4 Proving the two-dimensional bound . . . . . Bibliography
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Contents
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Preface
Summary The subject of this monograph can be described as the local properties of geometric Steiner minimal trees in finite-dimensional normed spaces. A Steiner minimal tree of a finite set of points is a shortest connected set interconnecting the points. For a quick introduction to this topic and an overview of all the results presented in this work, see Chapter 1. The relevant mathematical background knowledge needed to understand the results and their proofs are collected in Chapter 2. In Chapter 3 we introduce the Fermat-Torricelli problem, which is that of finding a point that minimizes the sum of distances to a finite set of given points. We only develop that part of the theory of Fermat-Torricelli points that is needed in later chapters. Steiner minimal trees in finite-dimensional normed spaces are introduced in Chapter 4, where the local Steiner problem is given an exact formulation. In Chapter 5 we solve the local Steiner problem for all two-dimensional spaces, and generalize this solution to a certain class of higher-dimensional spaces (CL spaces). The twodimensional solution is then applied to many specific norms in Chapter 6. Chapter 7 contains an abstract solution valid in any dimension, based on the subdifferential calculus. This solution is applied to two specific high-dimensional spaces in Chapter 8. In Chapter 9 we introduce an alternative approach to bounding the maximum degree of Steiner minimal trees from above, based on the illumination problem from combinatorial convexity. Finally, in Chapter 10 we consider the related k-Steiner minimal trees, which are shortest Steiner trees in which the number of Steiner points is restricted to be at most k.
Sources and co-authors All the results in this work. with the exception of some of the results in Chapter 10, have already been published or will appear soon. The new results in Chapter 3 are from my papers [97, 99]. Certain background results come from my paper [75], co-authored with Horst Martini and Gunter Weiß. The results of Chapter 5 have been published in my paper [99], with the exception of Sections 5.3 and 5.4, which will appear in my paper [73], co-authored with Horst Martini and my ex-PhD stu-
ix
Preface dent, P. Oloff de Wet. Of these, Section 5.3 is my own work, while Section 5.4 was done collaboratively. The examples in Chapter 6 originally appeared in my paper [99]. The results of Chapters 7 and 8 have been published in [101], and the results of Chapter 9 in [100]. The results of Chapter 10 have appeared in [98], with the exception of Section 10.1, which have not been published before.
Acknowledgements Many people have contributed directly or indirectly to the creation of this work. Firstly, and most importantly, I thank my coauthors Oloff de Wet [73], Horst Martini [73,75], and Gunter Weiß [75]. It was a pleasure working with them, and I hope that there will be more collaboration with them in the future. It would not be amiss to thank the referees of the papers [73, 75, 97–101]. Secondly, I thank the people who have stimulated my interest in geometric Steiner trees. Dietmar Cieslik’s pioneering papers and book [26] on Steiner minimal trees in Minkowski spaces was a great stimulus. Frank Morgan and his many students approached the topic in a very interesting way from the viewpoint of differential geometry [1,29,30,81,82]. Frank Morgan and Mark Conger supplied me with references and gave me encouragement. Horst Martini supported this project from the start, and made it possible for me to come to Germany. Nic van Rensburg drew my attention to the subdifferential calculus which plays a central role in this work. Thirdly, I thank the people who matter most in my life. Thank you, Annie, Karl and Leo, for everything.
Declaration I hereby declare that this monograph is my own work, based on the papers [73, 75, 97–101], of which only [73, 75] have been written with co-authors, as explained above.
Chemnitz,
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June 14, 2009
Theses
1. The local structure of a Steiner minimal tree in a finite-dimensional normed space is given an abstract characterization with the help of the subdifferential calculus, in terms of reduced Minkowski sums of the exposed faces of the unit ball of the dual space. 2. The characterization of the local structure of a Steiner minimal tree in finitedimensional normed spaces is used to determine the exact maximum degree of terminals and an asymptotic estimate of the maximum degree of Steiner points in the normed spaces that were conjectured by Cieslik to be extremal, and to give a partial solution to a conjecture of Conger on the maximum degrees in spaces with piecewise differentiable, elliptic norms. 3. The local structure of a Steiner minimal tree in a two-dimensional normed space is given a concrete characterization in terms of critical and absorbing angles. 4. The characterization of the local structure of a Steiner minimal tree in twodimensional normed spaces is used to obtain the maximum degree of Steiner points and of terminals in all these spaces, and to give a unified approach in proving various earlier known results. 5. The characterization of the local structure of terminals in a Steiner minimal tree in two-dimensional normed spaces in terms of absorbing angles is generalized to higher-dimensional, so-called Steiner antipodal normed spaces, of which CL spaces are a special case. 6. An asymptotic estimate is given to a quantitative version of Hadwiger’s illumination problem that was introduced by K. Bezdek, and a relationship to the maximum degree of terminals in Steiner minimal trees are found, which gives an asymptotic upper bound to the maximum degree of terminals, close to optimal. 7. A lower bound exponential in the dimension of the normed space is given to the maximum degree of a Steiner point in a k-Steiner minimal tree for all natural numbers k.
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Theses 8. The maximum degree of a Steiner point in a 1-Steiner minimal tree in a twodimensional normed space is at most 6, and it is exactly 6 only when the unit ball of the space is an affine regular hexagon.
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Chapter
1
A quick introduction and overview Various practical problems are modelled by Steiner minimal trees in finite-dimensional normed spaces. The study of these trees, known as the geometric Steiner problem, can be viewed as a certain one-dimensional Plateau problem. The local Steiner problem is the study of the singularities or the local structure of geometric Steiner minimal trees. There are many fascinating connections between this problem and various topics in mathematics, including the local theory of Banach spaces, the subdifferential calculus from convex analysis, the illumination problem from convex geometry, and extremal set theory from combinatorics. This introductory chapter gives an overview of this work.
1.1 The Steiner problem in normed spaces 1.1.1 The Euclidean Steiner problem We begin with a well-known motivational example. What is the shortest network that interconnects the three vertices of an equilateral triangle of edge length 1? A pair of edges of the triangle gives a tree T1 that interconnects the three vertices (Fig. 1.1(a)). This tree (a minimal spanning tree) has total length 2, but it is not the shortest T1
T2
(a) Minimal spanning tree
(b) Steiner minimal tree
Figure 1.1
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1 A quick introduction and overview T3
(a) Minimal spanning tree
T4
T5
(b) A shorter Steiner tree
(c) Steiner minimal tree
Figure 1.2
graph that connects the three vertices. √ Join the three vertices to the centroid of the triangle to obtain a tree T2 of length 3 (Fig. 1.1(b)). This new point is the Fermat-Torricelli point of the three vertices: the unique point whose sum of distances to the 3 vertices is a minimum. The tree T2 is indeed the shortest tree joining the original 3 vertices, but it has one more vertex. The new vertex is called a Steiner point, and T2 a Steiner minimal tree: a tree that is shortest among all those whose vertex set includes the original 3 points. Consider the four vertices of a square of edge length 1. Now a minimal spanning tree T4 has length 3 (Fig 1.2(a)). As before, the √tree T4 that joins all 4 vertices to the centre of the square is shorter, of length 2 2 (Fig 1.2(b)). Indeed, the centre is again the Fermat-Torricelli point of the four given points. However, there is an even shorter tree T5 interconnecting the 4 vertices, one with two Steiner points √ (Fig 1.2(c)). It has length 1 + 3, and is a Steiner minimal tree of the 4 vertices of the square. This example incidentally shows that Steiner minimal trees are in general not unique. Indeed, the rotation of the tree in 1.2 by 90◦ gives another Steiner minimal tree. In general, given any finite set S of terminals in some space, a Steiner tree is any tree whose vertex set is a subset of the space and contains S. A Steiner minimal tree (SMT) of S is then any shortest Steiner tree of S. The Steiner problem can now be described as the problem of finding a SMT of any given finite set of points in the space. Much has been said about the history of this problem. In particular we mention that Karl Menger already considered Steiner trees in general metric spaces [78], that Gilbert and Pollak [44] introduced the name Steiner minimal tree, since Courant and Robbins [31] referred to Steiner in their description of this problem. For a further historical information we refer the reader to [6, 26, 54, 59]. The algorithmic problem of finding a SMT of a given set of points is already NP-hard in the Euclidean plane [42], but there are polynomial time approximation schemes [4,5], as well as exact algorithms that are practically feasible at least for up to 2000 points [105].
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1.1 The Steiner problem in normed spaces
1.1.2 Other norms Here we give an overview of various normed spaces for which geometric Steiner minimal trees have been considered. This gives a sound motivation for studying SMTs in arbitrary (finite-dimensional) normed spaces. Another motivation for this generality is that many substantial results are valid for all norms, and can be proved using a very geometric approach. A finite-dimensional normed space (or Minkowski space) is a finite-dimensional real vector space V on which a norm has been defined, i.e., a function k·k : V → R such that k xk ≥ 0 for all x ∈ V with equality only for x = o, kλxk = |λ|k xk for all λ ∈ R and x ∈ V, and k x + yk ≤ k xk + kyk for all x, y ∈ V. Higher dimensional Euclidean spaces ¨ Mimura [79] and Jarn´ık and Kossler [56] were among the first to consider Steiner minimal trees in Euclidean spaces (see also [59]). High-dimensional Euclidean spaces have been used to model phylogenetic trees [13].
λ-geometry Motivated by application in engineering, Hanan [50] considered Steiner minimal trees in the rectilinear plane, where the norm is defined as k( x, y)k1 := | x | + |y|. This is the distance between two points if one is only allowed to use vertical and horizontal lines. This norm is the main norm used in VLSI design. Recently, more orientations have also been considered. For example, if we allow three orientations, each at 60◦ with respect to the other, we obtain the norm in which the unit ball is a regular hexagon. With four orientations, each at 45◦ degrees, we obtain the norm with unit ball a regular octagon. In general we may consider the so-called λgeometry, where the unit ball is the regular polygon with 2λ sides inscribed in the Euclidean circle [11, 14, 16]. For even more general considerations, see for example [108].
` p norm The ` p norm is very important in analysis, and has been considered in the Location Science literature to model distances between cities [17, 18, 33, 34]. It generalizes both the Euclidean norm and the `1 norm. For 1 ≤ p < ∞ it is defined on Rd as d
k( x1 , . . . , xd )k p :=
∑ | xi |
!1/p p
,
i =1
and for p = ∞ as
k( x1 , . . . , xd )k∞ := max{| xi | : i = 1, . . . , d}.
3
1 A quick introduction and overview Gradient-constrained norm The next example comes from the underground mining industry [12, 15]. Here the problem is to design a network of tunnels connecting a set of given underground locations where ore is concentrated. Because of limitations in the trucks used to haul the ore, the tunnels are not allowed to be too steep. Thus we constrain the gradient of each edge to be at most m, say. Apart from this constraint, the distance is Euclidean. This can be modelled as a norm: (p p x2 + y2 + z2 if |z| ≤ m x2 + y2 p k( x, y, z)k := √ 1 + m −2 | z | if |z| ≥ m x2 + y2 . The unit ball is the Euclidean unit ball with the north and south poles sliced off.
1.1.3 The one-dimensional Plateau problem One viewpoint, which originates in the theory of minimal surfaces, is to consider Steiner minimal trees as the lowest dimensional case of the general Plateau problem. This is the problem of finding a set of smallest measure that makes a given set more connected. The classical Plateau problem is to find a surface in R3 of smallest area which closes a given Jordan arc in 3-space. Here the study of singularities is very important, as they form an obstacle in getting a mathematical grip on minimizers [83]. Similarly, the Euclidean Steiner problem is to find a network of smallest length which connects a given set of points. These consideration lead Frank Morgan and his students to study the local structure of Steiner minimal trees for various norms. They were mostly interested in piecewise-C ∞ or uniformly convex norms (i.e., the curvature of the boundary of the unit ball is bounded away from 0—we wil call such norms elliptic). See [81, 82] for an exposition.
1.2 Looking at Steiner trees locally Our main topic is the Local Steiner problem: the study of the local structure of Steiner minimal trees in a finite-dimensional Minkowski space. We now make a more precise formulation. With local structure we mean the configuration of edges incident to a vertex. There are two types of vertices in a Steiner tree, the given points or terminals, and the Steiner points. The collection of edges emanating from a terminal is called a terminal configuration, and from a Steiner point a Steiner configuration.
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1.2 Looking at Steiner trees locally Clearly, if x is a Steiner point in a SMT with incident edges xvi , i = 1, . . . , n, then the star consisting of vertices { x, v1 , . . . , vn } and edges { xvi : i = 1, . . . , n} is a SMT of {v1 , . . . , vn }. A similar statement can be formulated for terminals. Thus we only have to consider stars. Without loss of generality, the centre x of the star may be taken to be the origin o. We may also assume that all the edges incident to o are of unit length: first scale the star so that all edges are larger than unit length; then it is clear that the star from o to all points on the edges at distance 1 from o must also be a SMT. Note that Steiner configurations form a subclass of terminal configurations. Indeed, if the star with edges ovi is a SMT of {v1 , . . . , vn }, then it is also a SMT of {o, v1 , . . . , vn }. The Local Steiner Problem. Given a finite-dimensional Minkowski space M, characterize all collections of unit vectors {v1 , . . . , vn } • that form a terminal configuration, i.e., such that the star from o to the vi is a SMT of {o, v1 , . . . , vn }, or • that form a Steiner configuration, i.e., such that the star from o to the vi is a SMT of {v1 , . . . , vn }. As an example, the solution in d-dimensional Euclidean space `2d (for any d ≥ 2) is the following: {v1 , . . . , vn } forms • a terminal configuration if, and only if, n ≤ 3 and all angles ^vi ovj ≥ 120◦ ; • a Steiner configuration if, and only if, n = 3 and all angles ^vi ovj = 120◦ . In other Minkowski spaces, our study of this problem utilizes various interesting fields of mathematics, in particular 1. Convexity: covering and illumination of convex bodies (Chapter 9, 2. Convex Analysis: the subdifferential calculus (Chapters 7 and 8), 3. Combinatorics: extremal finite set theory (Chapter 8), 4. Banach space theory and linear algebra: Cotype and 1-summing constants [96]. Thus this problem is not only of interest in itself. Instead, the connections to different fields of geometry and analysis enhances its importance. Before surveying the results contained in this work, we first discuss two conjectures that formed our main motivation.
1.2.1 Two guiding conjectures Let τ (M) [σ(M), resp.] denote the maximum degree of a terminal [Steiner point, resp.] in a SMT in the Minkowski space M, where the maximum is taken over all SMTs. These two values measure what may be called the local complexity of a SMT in M. A natural question is to determine these values for a given space M. This should be possible in principle once the local Steiner problem is solved for M. Observe that, since Steiner configurations are also terminal configurations, σ (M) ≤ τ (M).
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1 A quick introduction and overview Morgan’s conjecture Frank Morgan [81, 82] made the following conjecture:1 Conjecture 4.5 (Morgan [81, 82]). The maximum degree of a Steiner point in an SMT in any d-dimensional Minkowski space Md satisfies σ(Md ) ≤ 2d . It is not difficult to show that σ(`d∞ ) = 2d . Indeed, the star joining the origin to the 2d vertices of the unit ball is a SMT of the vertices. Thus the upper bound of 2d , if true, would be best possible. However, at least when d = 2, there are other norms also attaining 2d (Chapter 5), as for example the norm with the following unit ball [1]:
In Chapter 5 it is shown that this conjecture holds for d = 2, and all the 2-dimensional M2 with σ(M2 ) = 4 is characterized. Cieslik’s conjecture Dietmar Cieslik [21, 26] made a conjecture analogous to Morgan’s conjecture for τ (M). Conjecture 4.3 (Cieslik [21]). The maximum degree of an SMT in any d-dimensional Minkowski space Md satisfies τ (Md ) ≤ 2d+1 − 2, with equality if and only if Md is isometric to MdZ , the Minkowski space with unit ball conv([0, 1]d ∪ [−1, 0]d ). Cieslik [22] proved this conjecture for the case d = 2, where the unit ball is an affine regular hexagon. The unit ball of M3Z is affinely equivalent to the rhombic dodecahedron.
1 In this chapter, theorems and conjectures that are quoted from later chapters retain their numbering
from the chapter where they are proved. Thus Theorem 4.5 occurs in Section 4.3 of Chapter 4.
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1.3 Results and methods
1.3 Results and methods We now survey our main results on the local Steiner problem. We do not give a complete survey of all results by others. Instead, we emphasize the different techniques that we used in attacking this problem.
1.3.1 A general solution In Chapter 7 we present a general solution of the local Steiner problem for all Minkowski spaces, originally published in [101]. The solution involves the dual M∗ of the Minkowski space M, which is the vector space of all linear functionals on M, with norm k ϕk∗ := sup{ ϕ( x) : x ∈ M, k xk ≤ 1}. We say that a functional ϕ ∈ M∗ is a norming functional of x ∈ X if k ϕk∗ ≤ 1 and ϕ( x) = k xk. The collection of norming functionals of x ∈ M, denoted by ∂x := { ϕ ∈ M∗ : k ϕk∗ ≤ 1, ϕ( x) = k xk, is easily seen to equal the dual unit ball B∗ if x = o, and an exposed face of B∗ if x 6= o. In both cases ∂x is a closed convex subset of B∗ . We introduce a binary operation on the collection of closed, convex subsets of B∗ . Let C and D be two closed, convex subsets of B∗ . Their reduced Minkowski sum is defined to be C D = { ϕ + ψ : ϕ ∈ C, ψ ∈ D, k ϕ + ψk∗ ≤ 1}, i.e. C D is the usual Minkowski sum C + D intersected by the unit ball B∗ . Reduced Minkowski addition is clearly commutative but not associative. Thus there are different, inequivalent parenthesizations of n sets C1 , C2 , . . . , Cn , although some of them are equivalent after applying the commutative law. For example, (C1 C2 ) (C3 C4 ) is equivalent to (C4 C3 ) (C1 C2 ), but not to C1 (C2 C3 ) C4 . Theorem 7.2 (Nodes). Let N = { p0 , p1 , . . . , pk } be a set of points in a Minkowski space Mn . Then the star joining p0 to each pi , i ∈ [k ], is an SMT of N if and only if each parenthesization of {∂( pi − p0 ) : i ∈ [k ]} is non-empty. Theorem 7.3 (Steiner points). Let N = { p1 , . . . , pk } be a set of points in a Minkowski space Mn . Then the star joining p0 ∈ / N to each pi , i ∈ [k ], is an SMT of N if and only if each parenthesization of {∂( pi − p0 ) : i ∈ [k ]} with ∂( p1 − p0 ) on the outside, contains the zero functional o. For example, when n = 4 and { p1 , p2 , p3 , p4 } ⊂ M \ {o}: • the star joining o to { p1 , p2 , p3 , p4 } forms a terminal configuration if, and only, if
(∂pi ∂p j ) (∂pk ∂p` ) and ∂pi (∂p j (∂pk ∂p` )) are nonempty for all {i, j, k, `} = {1, 2, 3, 4} (15 conditions altogether);
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1 A quick introduction and overview • the star joining { p1 , p2 , p3 , p4 } forms a Steiner configuration if, and only if, the three sets
∂p1 (∂p2 (∂p3 ∂p4 )), ∂p1 (∂p3 (∂p2 ∂p4 )), ∂p1 (∂p4 (∂p2 ∂p3 )),
each contains o.
The proofs of Theorems 7.2 and 7.3 employ the subdifferential calculus of convex analysis, of which a thorough discussion is made in Chapter2. In particular, the subdifferential of the norm of M at x equals ∂x. If the norm is differentiable at x 6= o, ∂( x) = {∇k xk} is the singleton consisting of the gradient of the norm at x. Although Theorems 7.2 and 7.3 solve the local Steiner problem in a certain general sense, it is far from trivial to apply it for a specific norm. For instance, the number of inequivalent parenthesizations of n terms equals the product of the first n − 1 odd numbers: ∏in=−11 (2i − 1). However, we will apply this characterization for the space occurring in Cieslik’s conjecture (section 1.3.3), and for a certain space considered by Morgan (section 1.3.5). Also, in the smooth case reduced Minkowski addition is associative, which drastically simplifies the characterization and makes it much easier to apply (section 1.3.6).
1.3.2 The two-dimensional case In Chapter 5 we obtain a full solution to the local Steiner problem for two-dimensional Minkowski spaces (or Minkowski planes) in terms of so-called absorbing and critical angles. It is remarkable that two such simple notions are sufficient to give a characterization for all Minkowski planes. Although the solution uses some aspects of the general characterization theorems (Theorems 7.2 and 7.3), there are also a lot of ad hoc considerations which makes a generalization to higher dimensions impossible. An angle ^aob is absorbing if the point x = o minimizes the sum of the distances from the point x to a, o and b. Equivalently, { a, b} forms a terminal configuration. In the Euclidean plane an angle is obsorbing if, and only if, its angular measure is at least 120◦ . A Minkowski plane X is called an X-plane if there exist supplementary absorbing angles. For example, the planes with unit ball a square (`21 ), a regular hexagon (M2Z ), a regular octagon, a regular dodecagon (but no other regular polygon),
8
1.3 Results and methods
or even the following almost differentiable example:
The angle ^aob is critical if there exists a point c 6= o such that x = o minimizes the sum of the distances from x to a, b and c. In the Euclidean plane this is equivalent to having the angular measure equals 120◦ . See Chapter 5 for a slightly more detailed statement of the following theorem. Theorem 5.1. Consider any Minkowski plane M2 . 1. A star configuration in M2 is terminal if and only if all angles determined by the configuration are absorbing. 2. Degree four terminal configurations exist in M2 if and only if M2 is an X-plane. 3. Terminal configurations of degree five or six exist in M2 if and only if the plane is isometric to the 3-geometry. 4. Terminal configurations of degree more than 6 do not exist in M2 . 5. A star configuration of degree 3 in M2 is Steiner if and only if the configuration is floating FT, if and only if the configuration is not pointed and all angles are critical. 6. A star configuration of degree 4 in M2 is Steiner if and only if all angles are absorbing and some two angles are supplementary. 7. Steiner configurations of degree 4 exist in M2 if and only if the plane is an X-plane. 8. Steiner configurations of degree more than 4 do not exist in M2 . As a corollary, we obtain σ(M2 ) and τ (M2 ) for all planes:
9
1 A quick introduction and overview Plane M2 Non-X-plane X-planes except M2Z M2Z
σ(M2 ) 3 4 4
τ ( M2 ) 3 4 6
1.3.3 Cieslik’s conjecture and combinatorics In Chapter 8 we will use Theorem 7.2 to calculate τ (MdZ ) for the d-dimensional space Z d with unit ball conv(−1, 0)d ∪ [0, 1]d . Recall that part of Cieslik’s conjecture states that τ (MdZ ) = 2(2d − 1). When d = 2, the unit ball is an affine regular hexagon, where we have already seen that τ (M2Z ) = 6. d +2 Theorem 8.1. For all d ≥ 2, τ (MdZ ) = (b(d+ 2)/2c).
The unit ball, hence also the dual unit ball, of MdZ are polytopes, and it follows that the set of norming functionals of a point is also a polytope. It then possible to model reduced Minkowski addition combinatorially, and eventually the problem reduces to a problem in finite √ set theory. d +2 d Since (b(d+2)/2c) ∼ c2 / d, if 2(2d − 1) turned out to be the sharp upper bound for τ (Md ), the extreme situation is not attained by MdZ when d ≥ 3. This casts doubt on Cieslik’s conjecture. It is not clear which norms could be candidates for maximizing τ (Md ).
1.3.4 Quantitative illumination and covering In Chapter 9 we describe an approach to bounding τ (M) from above that is independent of Theorem 7.2. The famous illumination problem of Hadwiger and others is to show that any d-dimensional convex body B can be illuminated by at most 2d light sources. We say that a set P of points illuminates K if for each boundary point − → b of K there exists a point p ∈ P such that the ray pb intersects the interior of K after passing through b, i.e., {b + λ(b − p) : λ > 0} contains interior points of K. p2 P = { p1 , p2 , p3 }
K
b p1
10
p3
1.3 Results and methods There is an equivalent formulation in terms of covering. For any convex body K, the minimum number of light sources needed equals the minimum number of smaller homothets λK + v, λ < 1, of K that cover K. The light sources in the illumination problem can be arbitrarily far away. If K is centrally symmetric, without loss of generality K = −K, it defines a Minkowski space M with unit ball K and norm k·kK , which can then be used to measure how far away the light sources are. K. Bezdek [7] introduced the following parameter: I (K ) := inf{ ∑ k pkK : P illuminates K }. p∈ P
The connection with the local Steiner problem is the following theorem, proved in Chapter 9. Theorem 9.1. For any finite-dimensional Minkowski space M with unit ball B, σ (M) ≤ τ (M) ≤ I ( B ). Equality is possible in any dimension. For example, as will be seen later, σ(`1d ) = d ) = 2d . (Here Bd is the unit τ (`1d ) = I ( B1d ) = 2d, and σ(`d∞ ) = τ (`d∞ ) = I ( B∞ 1 d d ball of `1 , the d-dimensional cross polytope, and B∞ is the unit ball of `d∞ , the d-dimensional cube.) To apply this inequality, the immediate problem becomes that of bounding I ( X ) from above. In Chapter 9 we will connect Bezdek’s quantitative illumation problem to a quantitative covering problem, and then use results of Rogers on the covering of convex bodies to prove the following. Theorem 9.6. There exists a universal constant c > 0 such that for any d-dimensional Minkowski space Md with unit ball B, I ( B) < c2d d2 log d. Corollary 9.7. There exists a c > 0 such that for any d-dimensional Minkowski space Md , σ(Md ) ≤ τ (Md ) < c2d d2 log d. This gives a partial asymptotic solution to the conjectures of Morgan and Cieslik.
1.3.5 A conjecture of Conger Conger [30], a student of Morgan, studied the maximum degree of Steiner points when the norm is piecewise differentiable and elliptic, i.e., some small multiple of the Euclidean norm can be subtracted from the norm so that it stays a norm. Conjecture 4.6 Conger [82, p. 128]. For any d-dimensional Minkowski space Md with piecewise differentiable, elliptic norm, σ(Md ) ≤ 2d. A simple example of such a norm is the `1 norm with a small multiple of the Euclidean norm added. Using Theoremw 7.2 7.3 we prove the following in Chapter8. Theorem 8.12. Let λ√> 0, √ and let Md = (Rd , k·k1 + λk·k2 ). If λ ≤ 1 then σ(Md ) ≤ τ (Md ) ≤ 2d. If λ ≤ d/( d − 1) then τ (Md ) ≥ σ(Md ) ≥ 2d.
11
1 A quick introduction and overview
1.3.6 Differentiable spaces (including `dp ) and Banach space theory Although this work does not contain any further progress on the connection to the local theory of Banach spaces, we describe our previous work for the sake of completeness. When the norm is differentiable, any non-zero vector has a unique norming functional, and Theorem 7.2 reduces to the following two corollaries. The first characterizes Steiner points, and the second terminals. Corollary 7.4 (Lawlor and Morgan [63]). Let N = { p1 , . . . , pn } be a set of points, all different from the origin o, in a Minkowski space M with differentiable norm. Let ϕi be the norming functional of pi , i = 1, . . . , n. Then the star joining o to each pi is a SMT of N if and only if n
∑ ϕi = o
i =1
and for each subset I ⊆ {1, . . . , n},
k∑ ϕk∗ ≤ 1. i∈ I
Corollary 7.5 [96]. Let N = { p1 , . . . , pn } be a set of points, all different from the origin o, in a Minkowski space M with differentiable norm. Let ϕi be the norming functional of pi , i = 1, . . . , n. Then the star joining o to each pi is a SMT of {o} ∪ N if and only if for each subset I ⊆ {1, . . . , n}, k∑ ϕk∗ ≤ 1. i∈ I
Lawlor and Morgan used their characterization to show σ(Md ) ≤ d + 1 whenever the norm of Md is differentiable. Using a result on the invertibility of matrices, we strengthened this to τ (Md ) ≤ d + 1 [96]. We also used the notion of 1-summing constants to find another upper bound. The absolutely summing constant or the 1summing norm of (the identity operator on) a Minkowski space M is defined to be n o m m π1 (M) := inf c > 0 : ∀ x1 , . . . , xm ∈ X : ∑ k xi k ≤ c max k ∑ ε i xi k . i =1
ε i =±1
i =1
Theorem. For a Minkowski space M with differentiable norm, σ(M) ≤ τ (M) ≤ 2π1 (M∗ ).
√ It is known√that d ≤ π1 (Md ) ≤ d [57], which limits the upper bound found in this way to 2 d. Using the Khinchine inequalities we also obtained the following estimates for the `dp spaces, where 1 < p < ∞ [96]. Remarkably, the upper bounds are independent of the dimension. Theorem. If 2 ≤ p < ∞, then σ(`dp ) ≤ τ (`dp ) ≤ 7. If 1 < p < 2, then σ(`dp ) ≤ τ (`dp ) ≤ 2 p/( p−1) . In [96] more detailed estimates can be found for specific values of p.
12
1.4 Restricting the number of Steiner points
1.4 Restricting the number of Steiner points In Chapter 10 we consider the related, but much more difficult, k-Steiner minimal trees, where the number of Steiner points in the Steiner tree is bounded above by a natural number k. Such a shortest tree is called a k-Steiner minimal tree (k-SMT) of the given set of terminals. As before, we may define k-Steiner configurations and kterminal configurations to be the stars occurring in k-SMTs. We may also formulate the local k-Steiner problem as that of giving a characterization of all k-Steiner configurations and k-terminal configurations in a fixed Minkowski space. This turns out to be a much more difficult problem than the local Steiner problem. For example, the lengths of the edges in a star configuration matter. It is also not clear if k plays any role or not, locally. We may define σk (M) [τk (M)] to be the maximum degree of a Steiner point [resp. terminal] in a k-SMT in M. As Cieslik [25] showed, τk (M) equals the so-called Hadwiger number (or translative kissing number) of the unit ball B of M, i.e., the maximum number of pairwise non-overlapping translates of B that all touch B. The behavior of σk (M) is more complicated. It is clear that σk (M) ≤ τk (M). We show the following two bounds. The first is a lower bound valid for all spaces, exponential in the dimension, and independent of the specific space. Theorem 10.1. There exists c > 0 such that for any k ∈ N and d-dimensional Minkowski space Md , c σk (Md ) ≥ (4/3)d/4 . kd The second is an upper bound for 2-dimensional spaces M2 . As shown by Cieslik [24], σ1 (M2 ) ≤ 6 for all Minkowski planes. Theorem 10.5. For any Minkowski plane M2 , σ1 (M2 ) ≤ 5, except if M2 is isometric to the 3-geometry.
13
1 A quick introduction and overview
14
Chapter
2
Background knowledge, notation and definitions In this paper we collect the general mathematical background knowledge needed in the remainder of this work. Section 2.1 gives the basic definitions around signed sets, which is essential for getting a combinatorial grip on zonotopes (Section 2.5). In Section 2.2 we introduce our (abstract) approach to finite-dimensional vector spaces and their duals, as well as basic notions from convexity and topology. Section 2.4 introduces notions from Minkowski spaces, of which the most important is duality. After a discussion of zonotopes (Section 2.5), we end this chapter in Section 2.7 with a thorough discussion of the subdifferential calculus, which, as already hinted at in Chapter1, is crucial for most of what follows.
2.1 Signed sets Let [d] := {1, 2, . . . , d}, and let P [d] be the set of all subsets of [d]. Let | A| denote the number of elements in the finite set A. We use the following standard notation for signed sets (see [8]). A signed subset X of [d] is a vector X ∈ {0, ±1}d . We denote the ith component of X by X (i ) (reserving subscripts such as Xi for indices). The positive part of X is X + := {i : X (i ) = +1}, the negative part of X is X − := {i : X (i ) = −1}, the support of X is X := X + ∪ X − , and the zero set of X is X 0 := [d] \ X. We also write X = ( X + , X − ). Thus any two disjoint subsets A, B of [d] determine a signed set ( A, B). The natural partial ordering on signed sets X ≤ Y is defined by X + ⊆ Y + and X − ⊆ Y − . Two signed sets X, Y are conformal if X + ∩ Y − = ∅ = X − ∩ Y + . Denote the empty signed set (∅, ∅) by ∅.
2.2 Finite-dimensional vector spaces We denote the set of real numbers by R and the space of real d-tuples by Rd . The Kronecker delta is δij is defined to be 1 if i = j and 0 otherwise. It will be convenient to work with abstract finite-dimensional vector spaces instead of only Rd , as we
15
2 Background knowledge, notation and definitions have to consider subspaces and quotient spaces, for instance. Neverthless, we will also work with coordinates. We let V = V d denote a d-dimensional (abstract) vector space. We denote its elements in bold: a, x, etc. Denote its zero element or origin by o. We consider Rd to be the vector space of column vectors x = ( x(1), . . . , x(d))T , with the standard basis e1 , . . . , ed defined by ei ( j) = δij . The support of a vector x ∈ Rd is the signed set supp( x) := (supp+ ( x), supp− ( x)), where supp+ ( x) = {i : x(i ) > 0} and supp− ( x) = {i : x(i ) < 0}. Let x+ , x− be defined by x+ (i ) = max{ x(i ), 0} and x− (i ) = min{ x(i ), 0}. Consider an arbitrary subspace V ⊆ Rd and projection π : Rn → V. Define supp(V ) := {supp( x) : x ∈ V }. (This is the set of covectors of the oriented matroid of the vector configuration {π (ei ) : i = 1, . . . , d} in V; cf. [8, §2.2].) The dual V ∗ of an abstract vector space V is the set of all linear functionals ϕ : V → R. We denote the zero functional by o. A hyperplane in V is any set of the form ϕ−1 (λ) = { x ∈ V : ϕ( x) = λ}, where ϕ ∈ V ∗ \ {o } and λ ∈ R. We identify the dual (Rd )∗ with the set of row vectors ϕ = ( ϕ(1), . . . , ϕ(n)), with standard basis ei = eT i . Again consider an arbitrary subspace V ⊆ Rd , and now let π : Rd → V be the orthogonal projection. The quotient space Rd / ker π can then be identified with V. The dual (Rd / ker π )∗ is in a canonical way isomorphic to the annihilator of ker π, i.e., to the subspace { ϕ : ker π ⊆ ker ϕ} of (Rd )∗ . We accordingly identify this space with V ∗ . It is easily seen that with these identifications supp(V ) = supp(V ∗ ) (as long as π is orthogonal). For any two abstract vector spaces V and W, (V × W )∗ is canonically isomorphic to V ∗ × W ∗ , since a functional ϕ ∈ (V × W )∗ may be decomposed uniquely as ϕ( x, y) = ψ( x) + χ(y) for some ψ ∈ V ∗ and χ ∈ V ∗ ; in fact, ψ( x) = ϕ( x, o) and χ(y) = ϕ(o, y). Conversely, given any ψ ∈ V ∗ , χ ∈ W ∗ , their sum ψ( x) + χ(y) defines a functional on V × W which we denote (ψ, χ).
2.3 Convexity, topology, and measure For more on convex geometry see [46, 92]. If A and B are subsets of a vector space V, we define their Minkowski sum to be A + B := { a + b : a ∈ A, b ∈ B}. We also define − A := {o} − A, λA := {λa : a ∈ A}, and A − B := A + (− B). The affine span of a subset S ⊆ V is the set of all affine combinations of elements of S: n
n
i =1
i =1
aff S := { ∑ λi si : si ∈ S, λ ∈ R, ∑ λi = 1}, and the convex hull of S is the set of all convex combinations of elements of S: n
n
i =1
i =1
conv S := { ∑ λi si : si ∈ S, λ ≥ 0, ∑ λi = 1}. The line segment with endpoints x and y in V is xy := conv{ x, y} = {αx + (1 − α)y : 0 ≤ α ≤ 1},
16
2.4 Minkowski spaces and the ray with origin x passing through y is
− → xy := {αx + (1 − α)y : α ≤ 1}. − → − → An angle ^xyz in a vector space is the convex cone bounded by two rays yx and yz emanating from the same point y: ^xyz := {αx + βy + γz : α, γ ≥ 0, α + β + γ = 1}. (We allow half planes — in this case we take the half plane on the left if we pass from x to z.) The interior, closure, and boundary of a subset S ⊂ V is denoted by int S, cl S, and bd S, respectively. The relative interior of S is the interior of S in the subspace topology of aff S. A set C ⊆ V is convex if xy ⊆ C for all xy. In particular, convex hulls and affine spans are alwys convex. The set C is a convex body if it is convex, compact, and has non-empty interior. A convex body C is centered if C = −C. The central symmetral of a convex body C is 12 (C − C ), which is a centered convex body. We will use the following version of the (weak) separation theorem [86, Theorem 11.3]. Separation Theorem. Let C1 and C2 be non-empty convex sets with disjoint relative interiors in a finite-dimensional real vector space. Then there exists a hyperplane H such that C1 and C2 are in opposite closed half spaces bounded by H. As usual, a function f : V → R is convex if f (λx + (1 − λ)y) ≤ λ f ( x) + (1 − λ) f (y) for all x, y ∈ V and 0 ≤ λ ≤ 1. Convex functions on V are also continuous on V. Denote the d-dimensional Lebesgue measure, or volume of a measurable subset S ⊆ Rd by vol(S). Note that a convex body K ⊂ Rd is always measurable, has finite volume, and vol(λK ) = |λ|d vol(K ) for any λ ∈ R. The translative covering density of a convex body K in a finite-dimensional vector space V is defined as follows. A covering of V by translates of K is a collection of translates C = {vi + K : i ∈ I } such S that i∈ I vi + K = V. The (lower) density of the covering is defined to be ∑ vol(λK ∩ i∈ I vi + K ) ϑ (C) := lim inf i∈ I . vol(λK ) λ→∞ S
It is clear that the ratio is independent of the basis chosen (which is needed to calculate vol). The translative covering density of K is then defined as ϑ (K ) := inf ϑ (C), where the infimum is taken over all coverings of V by K. It is clear that ϑ (K ) ≥ 1 for all convex bodies K. According to the Rogers-Shephard inequality [88], vol(K − K )/ vol(K ) ≤ (2d d ) (and of course if K is centered then vol( K − K ) = d vol(2K ) = 2 vol(K )).
2.4 Minkowski spaces For more on Minkowski geometry, see [71, 74, 104]. A Minkowski space is a finitedimensional vector space V together with a with a norm k·k : V → R which satisfies
17
2 Background knowledge, notation and definitions
k xk ≥ 0, k xk = 0 if and only if x = o, kλxk = |λ|k xk, and most importantly, the triangle inequality k x + yk ≤ k xk + kyk). Denote a Minkowski space by M = (V, k·k), or Md if it is d-dimensional. When d = 2 we speak of a Minkowski plane. The unit ball B = B(M) = { x ∈ V : k xk ≤ 1} determines the norm uniquely, and we also write M = (V, B). In fact, any centered convex body B defines a norm for which B is the unit ball: k xk B = inf{λ−1 : λx ∈ B}. By the Mazur-Ulam Theorem, any two Minkowski spaces are isometric iff their unit balls are affinely equivalent, i.e., if there exists a linear mapping from one unit ball onto the other. The unit sphere S = S(M) is the boundary of the unit ball: S := bd B. The normalization of x 6= o is 1 xb := x. k xk Let B( a, r ) = { x : k x − ak ≤ r } denote the closed ball with center a and radius r. The dual of M = (V, k·k) is M∗ = (V ∗ , k·k∗ ), with the dual norm defined by
k ϕk∗ := max{ ϕ( x) : k xk = 1}. Denote the dual unit ball by B∗ and the dual unit sphere by S∗ . If M is identified with Rd , and M∗ = (Rd )∗ also identified with Rd , then ϕ( x) = h ϕ, xi, and B∗ is the polar body of B: B∗ = {y ∈ Rd : h x, yi ≤ 1 ∀ x ∈ B}. It is well-known that the dual of the dual is again the original space. The following lemma is standard. Lemma 2.1. Let k·k a and k·kb be any norms on V. Then the dual of M = (V, k·k a + k·kb ) is isometrically isomorphic to (V ∗ , Ba∗ + Bb∗ ), where Ba∗ and Bb∗ are the unit balls of the dual norms k·k∗a and k·k∗b . The ` p norm on Rd is defined by d
k( x(1), . . . , x(d))k p = ( ∑ | x(i )| p )1/p i =1
if p ≥ 1, and
k( x(1), . . . , x(d))k∞ = max{ x(i ) : i ∈ [d]}. The space `dp is (Rd , k·k p ) = (Rd , Bdp ), where we denote the ` p unit ball by Bdp . The d , and the d-dimensional cross polytope is Bd . The spaces `d and `d are d-cube is B∞ ∞ 1 1 dual. When 1/p + 1/q = 1 then `dp and `dq are also dual. A space is Euclidean if its p norm comes from some inner product: k xk = h x, xi. An ellipsoid is the unit ball of any Euclidean space. Any Euclidean space is isometric to `2d . The λ-geometry, introduced in [91], is the normed plane with a regular 2λ-gon as unit ball. The 2-geometry is also called the rectilinear plane and is isometric to `21 and `2∞ .
18
2.4 Minkowski spaces Any x ∈ M has a norming functional, i.e. a functional ϕ such that k ϕk∗ ≤ 1 and ϕ( x) = k xk. For x 6= o this can be seen by applying the Separation Theorem to B(M) and xb, which shows that any boundary point of B has a supporting hyperplane. We denote the set of norming functionals of x by ∂x. In particular, ∂o = B∗ . When x 6= o, ∂x is an exposed face of B∗ , i.e., the intersection of B∗ by some supporting hyperplane, in this case B∗ ∩ { ϕ : ϕ( x) = k xk}. In particular, ∂x ⊂ S∗ . More generally, the exposed face of the unit ball B defined by a unit functional ϕ ∈ M∗ is denoted by [ ϕ ∗ ] : = { x ∈ B : ϕ ( x ) = 1}, and the exposed face of B∗ defined by the unit vector u ∈ Md is denoted by [u]∗ . If B is a polytope then all faces are exposed, and each face F of B corresponds to a face F ∗ of B∗ as follows: F ∗ := { ϕ ∈ B∗ : ϕ( x) = 1 for all x ∈ F }. A norm is differentiable at x if and only if ∂x is a singleton for each x 6= o, and then ∂x = {∇k xk} [86, Theorem 25.1]. If the norm is differentiable at each x 6= o, then it is in fact C1 , i.e., the gradient mapping ∇ : V \ {o} → S∗ is continuous. Note that since ∇kλxk = ∇k xk for all x 6= o, λ > 0, all information about this mapping is already contained in its restriction to the unit sphere S. We say that M is a C1 space if the norm is differentiable on V \ {o}. In Functional Analysis, such a space is usually called a smooth space. A Minkowski space M (as well as its unit ball) is strictly convex if for all linearly independent x, y ∈ M we have k x + yk < k xk + kyk. Equivalently, the unit sphere does not contain a non-trivial segment xy, x 6= y. A norm is strictly convex if and only if the dual norm is differentiable. We now discuss a subtle strengthening of both differentiability and strict convexity, related to curvature. A norm is called C1,1 if it is differentiable and the gradient mapping ∇k·k is locally Lipschitz on V \ {o}: for each v ∈ V \ {o} there is a neighborhood U of v and a K > 0 such that for any x, y ∈ U,
k∇k xk − ∇kykk∗ ≤ K k x − yk. Equivalently, the restriction of this mapping to the unit sphere is Lipschitz. If a C1,1 norm is C2 , then the curvature of the unit sphere is bounded from above. A Minkowski space M (as well as its norm k·k and unit ball B) is elliptic if k·k − εk·k E is still a norm for some ε > 0 and Euclidean norm k·k E . This is independent of the choice of Euclidean structure, since for any two Euclidean norms k·k E and k·ke there exists ε > 0 such that k·k E − εk·ke is still a norm. (This follows essentially from the fact that when a sufficiently small multiple of the identity matrix is subtracted from a positive definite matrix, the result is still positive definite.) If an elliptic norm is C2 , then the curvature of the unit sphere is bounded away from 0. A norm is elliptic if and only if the dual norm is C1,1 [80, §3]. From this and Lemma 2.1 it follows that a norm is C1,1 if and only if its unit ball can be written as B = C + εBE for some centered convex body C, some ε > 0, and some ellipsoid BE . Note that in Functional Analysis a differentiable norm is also called a smooth norm, while in differential geometry the word smooth is usually reserved for C ∞ .
19
2 Background knowledge, notation and definitions In differential geometry the unit ball of an elliptic norm is also called uniformly convex, which again has another meaning in Banach space geometry. We therefore avoid the terms smooth and uniformly convex in this paper, and stick to the terms differentiable and elliptic. We now discuss the consequences of non-strict convexity. A long segment is a segment xy on the unit sphere of length k x − yk > 1. Lemma 2.2. A segment xy on the unit sphere S of M has length at most 2, with equality if and only if B ∩ aff{o, x, y} is a parallelogram. Lemma 2.3. Let a, b, c be unit vectors in M, with b and o on opposite sides of the line aff{ a, c}. Then k a − bk ≤ k a − ck. If furthermore k a − bk = k a − ck = 1, then b, c, b − a, c − a are collinear unit vectors, and b(c − a) is a long segment on the unit circle, parallel to aff o, a. b c a
c−a
o
Lemma 2.4. Let x and y be linearly independent vectors in M. Then k x + yk = k xk + kyk if and only if xbyb is a segment on the unit sphere. Proof. Since x and y are linearly independent, x 6= o, y 6= o, x + y 6= o. If k x + yk = k xk + kyk, then k xk kyk x[ +y = xb + yb, k x + yk k x + yk which shows that x[ + y ∈ ( xb, yb). Therefore, [ xb, yb] consists only of unit vectors. Conversely, it follows from the convex combination
k xk kyk 1 ( x + y) = xb + yb k xk + kyk k xk + kyk k xk + kyk that
1 ( x + y) k xk+kyk
is a unit vector; hence k x + yk = k xk + kyk.
Lemma 2.5. Let x, y 6= o be given in a M such that k x − yk ≥ ko − yk ≥ ko − xk. Then xbyb ≥ 1, with equality iff either 1. k x − yk = ko − xk = ko − yk, or 2. k x − yk = ko − yk > ko − xk and yb(yb − xb) is a segment on the unit sphere. Proof. The triangle inequality gives
k
20
1 1 1 1 1 1 ( x − y)k ≤ k ( x − y) + ( − )yk + k−( − )yk k xk k xk k xk kyk k xk kyk 1 1 = k xb − ybk + k( − ) y k. kyk k xk
(2.1)
2.5 Zonotopes as unit balls Therefore, 1 1 1 k x − yk − ( − )kyk k xk k xk kyk 1 (k x − yk − ko − yk) + 1 = k xk ≥ 1.
xbyb ≥
If xbyb = 1, then yb − xb is a unit vector, k x − yk = ko − yk, and there is equality in the triangle inequality (2.1). If furthermore kyk > k xk, then Lemma 2.4 gives that yb(yb − xb) is a segment on the unit sphere. We denote the d-segment from x to y by
[ xy]d := {z ∈ X : k x − zk + kz − yk = k x − yk}. A metric ray is a subset of M that is isometric to any ray, and a metric line is a subset of M isometric to any line. If W is a subspace of M = (V, k·k), then the quotient space M/W is the space V/W with norm k x + W k = inf{k x + wk : w ∈ W }. If V is a subspace of M = (Rn , k·k), a concrete presentation of M/V can be obtained by using the orthogonal projection π : Rn → V ⊥ . Then M/V is isometrically isomorphic to (V ⊥ , π ( B)), i.e., we project the unit ball B orthogonally onto V ⊥ to obtain the norm. The dual (M/V )∗ is as before the annihilator of ker π = V, and the dual norm is a restriction of the dual norm k·k∗ of M∗ , i.e., the dual is a subspace of M∗ , with the embedding given by the adjoint π ∗ of the projection π.
2.5 Zonotopes as unit balls We will have two occasions to use zonotopes. In Chapter 3 we need the fact that any centered convex polygon in the plane is a zonotope. In Chapter 8 we use the fact that the unit ball of Cieslik’s space MdZ is a zonotope. We now need the notation of signed sets (Section 2.1).
2.5.1 General combinatorial description A zonotope Z in a d-dimensional vector space V is a Minkowski sum of segments Z = x1 y1 + x2 y2 + · · · + xn yn where x1 , . . . , xn , y1 , . . . , yn ∈ V. Equivalently, a zonotope is the projection of an d-cube onto a subspace. The zonotope determined by a subspace V of Rd is the d ) of the d-cube onto V. As explained in Section 2.4, orthogonal projection Z = π ( B∞ (V, Z ) is isometric to the quotient `d∞ / ker π, and its dual (V ∗ , Z ∗ ) is isometric to the subspace im π ∗ of `1d . As before, we identify V ∗ with this subspace. Then Z ∗ = V ∗ ∩ B1d . We need a description of the faces of Z ∗ . Each proper face of Z ∗ is the
21
2 Background knowledge, notation and definitions
Figure 2.1: Two representations of the rhombic dodecahedron intersection of V ∗ with a proper face of the cross polytope B1d , i.e., there is one for each signed set X ∈ supp(V ∗ ), namely F 0 ( X ) = { ϕ ∈ V ∗ : k ϕk1 = 1, supp( ϕ) ≤ X }, of dimension | X | − 1. Since X 7→ F 0 ( X ) is order-preserving, F 0 ( X ) is a facet if and only if X is a maximal covector in supp(V ∗ ). Two faces F 0 ( X ) and F 0 (Y ) belong to a common facet if and only if X and Y are conformal. We mention that the corresponding non-empty faces of Z are indexed by the same set supp(V ) = supp(V ∗ ). Each such face equals some F(X ) =
∑
i∈X +
π ( ei ) −
∑
π (e j ) +
j∈ X −
∑ [−π (ek , π (ek )],
k∈ X0
of dimension F ( X ) = | X 0 | (see [8, §2.2]). Since the mapping X 7→ F ( X ) is orderreversing, F ( X ) is a vertex if, and only if, X is a maximal covector in supp(V ∗ ), while two faces F ( X ) and F (Y ) intersect if and only if X and Y are conformal.
2.5.2 Cieslik’s norm We now consider the Minkowski space MdZ considered by Cieslik. The unit ball of M2Z is an affine regular hexagon, and of M3Z an affine rhombic dodecahedron. As formulated in Cieslik’s conjecture in Chapter 1, we took the unit ball of MdZ to be conv([0, 1]d ∪ [−1, 0]d ). Then its norm is
k( x(1), . . . , x(d))k = max x(i ) − min x(i ). x(i )≥0
x(i )≤0
See Figure 2.1 for two affine representations, one as in the conjecture, and the other with all faces congruent. We now want to represent MdZ in the second, more symmetric, way. Consider the hyperplane H = { x ∈ Rd +1 :
d +1
∑ x ( i ) = 0}
i =1
22
2.5 Zonotopes as unit balls of Rd+1 , and let π be the orthogonal projection of Rd+1 onto H along the vector j := (1, 1, . . . , 1)T . We now formally define MdZ to be the quotient `d∞+1 / ker π, d+1 ). It which we identify with the d-dimensional space H with unit ball Zd := π ( B∞ is easily seen that k xk Z = 21 ( max x(i ) − min x(i )). i ∈[d+1]
i ∈[d+1]
The dual space MCd := (MdZ )∗ is the subspace H ∗ = { ϕ ∈ (Rd +1 ) ∗ :
d +1
∑ ϕ ( i ) = 0}
i =1
of `1d+1 , with support
Zd := supp( H ∗ ) = { X ∈ {0, ±1}d+1 : X + 6= ∅, X − 6= ∅} ∪ {∅}. Note that F 0 (∅) = ∅, and all other F 0 ( X ) are non-empty proper faces of Zd∗ . The facets of Zd∗ are all the F 0 ( X ) with X ∈ {±1}d+1 , X ± 6= ∅. In general, F 0 ( X ) = { ϕ ∈ (Rd +1 ) ∗ :
∑+ ϕ(i) = 21 , ∑− ϕ(i) = − 21 ,
i∈X
supp( ϕ) ≤ X }.
i∈X
Also define Zd := Zd ∪ {1}, with 1 a symbol corresponding to the improper face Zd∗ . We consider 1 ≥ X for all X ∈ Zd . Thus (Zd , ≤) is the face lattice of Zd∗ . It is easy to see that Zd∗ is the difference body of the equilateral d-simplex ∆d := conv{ei : i ∈ [d + 1]}. Thus Z2∗ is a regular hexagon, and Z3∗ is a cuboctahedron. To show that Zd is affinely equivalent to conv([0, 1]d ∪ [−1, 0]d ), let π 0 be the (non-orthogonal) projection of Rd+1 onto the hyperplane H 0 = { x ∈ Rd +1 : x ( d + 1 ) = 0 } , again along the vector j. Then π 0 | H and π | H 0 are inverses of each other taking Zd to π 0 (C d+1 ) and vice versa. Identifying H 0 with Rd in the obvious way, it is easily seen that π 0 (C d+1 ) = conv([0, 2]d ∪ [−2, 0]d ).
2.5.3 Two-dimensional zonotopes In the two-dimensional situation, note that any centered two-dimensional polytope (or polygon) is always a zonotope [110, Example 7.14]. In particular, if x1 , . . . , x n , − x1 , . . . , − x n are the consecutive edges of a the centered polygon P symmetric around 0, then P can be written as the Minkowski sum n
P=
1
1
∑ conv{ 2 (xi+1 − xi ), 2 (xi − xi+1 )},
(2.2)
i =1
where we set xn+1 := − x1 .
23
2 Background knowledge, notation and definitions
2.6 Hanner polytopes and CL spaces A Minkowski space is a CL space if for every maximal proper face F of the unit ball B we have B = conv( F ∪ (− F )). It is easily seen from finite dimensionality that the unit ball of a CL space is a polytope. CL spaces were introduced by R. E. Fullerton (see [84]), although the notion has been studied before by Hanner [51], who proved that the unit balls of CL spaces are {0, 1}-polytopes. McGregor [76] showed that CL spaces are exactly those spaces with numerical index 1. Hanner [51] identified an important subclass of CL spaces, namely those that can be built up from the one-dimensional space R using `1 -sums and `∞ -sums. For two Minkowski spaces M and N of dimension d and e we define their `1 -sum M ⊕1 N and `∞ -sum M ⊕∞ N to be the Minkowski spaces on Rd+e with norms k( x, y)k1 = k x k + kyk and k( x, y)k∞ = max{k x k, kyk}. Note that the unit ball of M ⊕1 N is the convex hull of the unit ball of M when embedded as M ⊕ {o } and the unit ball of N when embedded as {o } ⊕ N. The unit ball of M ⊕∞ N is the Cartesian product of the unit balls of M and N. The unit balls of these spaces are called Hanner polytopes. We thus introduce the name Hanner space for these spaces. Proposition 2.6. All Hanner spaces are CL spaces. The dual of a Hanner space is again a Hanner space. The dual of a CL space is again a CL space. Proof. For the first statement see e.g. [84]. It is clear from the definition of Hanner space that its dual is also a Hanner space. For the last statement see [76]. For more information see [51–53, 84].
2.7 Subdifferential calculus In this section we summarize the basic results of the subdifferential calculus needed in this paper. For more on subdifferential calculus, see [86, §23] and [109]. We only consider convex functions defined everywhere in finite-dimensional spaces, so many of the arguments simplify. Although these results are all well-known in convex analysis, they are not easy to locate in the same reference, and the proofs given are usually based on a variety of other more general theorems. For the convenience of the reader we give direct proofs of these lemmas. Let V be any finite-dimensional vector space with dual V ∗ . A functional ϕ ∈ V ∗ is a subgradient of a convex function f : V → R at the point a if for all x ∈ V, f ( x ) − f ( a ) ≥ ϕ ( x − a ). In particular, o ∈ V ∗ is a subgradient of f at a if and only if f attains its minimum value at a. The subdifferential of f at a, denoted by ∂ f ( a), is the set of all subgradients of f at a. Note that for any ϕ ∈ V ∗ and c ∈ R, ∂( ϕ + c)( a) = { ϕ} for all a ∈ V. Lemma 2.7. The subdifferential of a convex function f : V → R at any a ∈ V is nonempty, compact, and convex.
24
2.7 Subdifferential calculus See [86, Theorem 24.7]. Proof. Since ∂ f ( a) = x∈V { ϕ ∈ V ∗ : ϕ( x − a) ≤ f ( x) − f ( a)}, which is an intersection of closed half spaces in V ∗ , we have that ∂ f ( a) is closed and convex. If ∂ f ( a) is unbounded, a compactness argument shows that it contains a ray { ϕ0 + λϕ1 : λ ≥ 0}. Therefore, f ( x) − f ( a) ≥ ϕ0 ( x − a) + λϕ1 ( x − a) for all x ∈ V and λ ≥ 0. It follows that ϕ1 ( x − a) ≤ 0 for all x ∈ V, a contradiction. To show that ∂ f ( a) 6= ∅ we use the separation theorem to obtain a hyperplane that separates the point ( a, f ( a)) and the epigraph of f T
{( x, y) : x ∈ V, y ≥ f ( x)} ⊆ V × R. Thus there exists a non-zero functional ( ϕ, r ) ∈ (V × R)∗ = V ∗ × R such that ( ϕ, r )( x, y) ≥ ( ϕ, r )( a, f ( a)) for all x ∈ V and y ≥ f ( x). Then ϕ( x − a) ≥ r ( f ( a) − y), which gives that r ≥ 0, since y can be arbitrarily large. If r = 0 this also gives ϕ = o, contradicting ( ϕ, r ) 6= (o, 0). Therefore, r > 0, and − 1r ϕ( x − a) ≤ f ( x) − f ( a) for all x ∈ V, giving − 1r ϕ ∈ ∂ f ( a). Lemma 2.8 (Moreau-Rockafellar Theorem). Let f i : V → R, i = 1, . . . , k be convex functions. Then k
∂( ∑ f i )( a) = i =1
k
∑ ∂ f i ( a ),
i =1
where the sum on the right is Minkowski addition. See [86, Theorem 23.8] and [109, Theorem 2.8.7]. Proof. The “⊇”-inclusion is straightforward, but the “⊆”-inclusion needs the separation theorem (see the second proof on p. 224 of [86]). We prove the case k = 2, with the general case following by induction. Let f and g be convex functions on V and let ϕ ∈ ∂( f + g)( a). Then f ( x) − f ( a) + g( x) − g( a) ≥ ϕ( x − a) for all x ∈ V.
(2.3)
The sets C1 = {( x, r ) ∈ V × R : f ( x) − f ( a) − ϕ( x − a) ≤ r } and C2 = {( x, r ) ∈ V × R : g( x) − g( a) ≤ −r } are both convex and closed with non-empty interior, and by (2.3) their interiors are disjoint. They have a common point ( a, 0). By the separation theorem there exists a non-zero (ψ, s) ∈ (V × R)∗ = V ∗ × R such that (ψ, s)( x, r ) ≥ (ψ, s)( a, 0) for all ( x, r ) ∈ C1 and (ψ, s)( x, r ) ≤ (ψ, s)( a, 0) for all ( x, r ) ∈ C2 . This gives that for all x ∈ V, ψ( x − a) ≥ −sr for all r ≥ f ( x) − f ( a) − ϕ( x − a), (2.4) and ψ( x − a) ≤ −sr for all r ≤ g( a) − g( x).
25
2 Background knowledge, notation and definitions Since r can be arbitrarily large in (2.4), s ≥ 0. If s = 0, then ψ( x − a) = 0 for all x ∈ V, contradicting (ψ, s) 6= (o, 0). Therefore, s > 0, and we obtain
− 1s ψ( x − a) ≤ f ( x) − f ( a) − ϕ( x − a) and 1 s ψ( x
− a ) ≤ g ( x ) − g ( a ).
This gives ϕ − 1s ψ ∈ ∂ f ( a) and 1s ψ ∈ ∂g( a). Adding, we obtain ϕ ∈ ∂ f ( a) + ∂g( a), as required. Lemma 2.9. Let f i : Vi → R, i = 1, . . . , k be convex, and define f : V1 × · · · × Vk → R by f ( x1 , . . . , xk ) = ∑ik=1 f i ( xi ). Then the subdifferential of the convex function f is the Cartesian product k
∂ f ( a1 , . . . , a k ) =
∏ ∂ fi (ai ) ⊆ V1∗ × · · · × Vk∗ . i =1
See [109, Corollary 2.4.5]. Proof. By induction it is sufficient to consider the case k = 2. Let f : V → R and g : W → R be convex, define F : V × W → R by F ( x, y) = f ( x) + g(y), and choose a ∈ V, b ∈ W. First choose ϕ ∈ ∂ f ( a) and ψ ∈ ∂g(b). Then f ( x) − f ( a) ≥ ϕ( x − a) for all a ∈ V, and g(y) − g(b) ≥ ψ(y − b) for all b ∈ W. Adding, we obtain F ( x, y) − F ( a, b) ≥ ( ϕ, ψ)( x, y) − ( ϕ, ψ)( a, b), which gives ( ϕ, ψ) ∈ ∂F ( a, b). Now choose ( ϕ, ψ) ∈ ∂F ( a, b). This means that for all x ∈ V and y ∈ W, ϕ ( x − a ) + ψ ( y − b ) ≤ f ( x ) − f ( a ) + g ( y ) − g ( b ). Setting y = b we obtain ϕ ∈ ∂ f ( a), and setting x = a gives ψ ∈ ∂g(b). We have already defined the set of norming functionals ∂x of any x in a Minkowski space M (Section 2.2). We now explain the reason for this notation. Lemma 2.10. The subdifferential of the norm of M at a is given by ∂k ak = ∂a. See [109, Corollary 2.4.16]. Proof. We have that ϕ ∈ B(M∗ ) if and only if ϕ( x) ≤ k xk for all x ∈ M, which is equivalent to ϕ ∈ ∂kok. This takes care of the case a = o. If a 6= o, then for any ϕ ∈ ∂a and any x ∈ M, ϕ ( x − a ) = ϕ ( x ) − k a k ≤ k x k − k a k, hence ϕ ∈ ∂k ak. Conversely, suppose
k xk − k ak ≥ ϕ( x − a) for all x ∈ M.
(2.5)
Setting x = o in (2.5) we obtain ϕ( a) ≥ k ak, giving k ϕk∗ ≥ 1. Setting x = 2a in (2.5) we obtain k ak ≥ ϕ( a), giving ϕ( a) = k ak. Now (2.5) gives k xk ≥ ϕ( x) for all x ∈ M, implying k ϕk∗ = 1. It follows that ϕ is a norming functional of a.
26
2.7 Subdifferential calculus The distance function ρ( x, y) := k x − yk is easily seen to be convex on M × M by the triangle inequality. Lemma 2.11. For any a, b ∈ M, ∂ρ( a, b) = {( ϕ, − ϕ) : ϕ ∈ ∂( a − b)} ⊆ M∗ × M∗ . Proof. Let ϕ ∈ ∂( a − b). By Lemma 2.10 we have for any x, y ∈ M,
k x − yk − k a − bk ≥ ϕ( x − y − a + b) = ϕ( x − a) − ϕ(y − b) = ( ϕ, − ϕ)( x − a, y − b), which gives ( ϕ, − ϕ) ∈ ∂ρ( a, b). Conversely, let ( ϕ, ψ) ∈ ∂ρ( a, b). Then, in particular, for any x ∈ M, ρ( x, x) − ρ( a, b) ≥ ϕ( x − a) + ψ( x − b), i.e., ( ϕ + ψ)( x) ≤ ϕ( a) + ψ(b) − k a − bk. Since the right-hand side is independent of x, we obtain ϕ + ψ = o.
27
2 Background knowledge, notation and definitions
28
Chapter
3
Fermat-Torricelli points and special angles In this chapter we introduce the notion of a Fermat-Torricelli point (FT point) of a collection of points. This is important for the local Steiner problem, since a Steiner point in a SMT is a FT point of its neighbors, while a terminal in a SMT is a FT point of the set consisting of its neighbors and itself. We develop the theory as far as needed for later chapters. The results in this chapter have their origin in [75, 97, 99]. For a much more through investigation see [75]. The notions of absorbing and critical angles are very important. Although the results about these angles are essentially two-dimensional, and will be applied in Chapter 5, at the end of that chapter it will be shown that these angles also have a use in certain higher dimensional spaces. In the first section we introduce the Fermat-Torricelli point and study its basic properties. In Section 3.2 we use the notion of zonotopes in two dimensions to prove a result for Minkowski planes that generalizes a Euclidean result [60]. In Section 3.3 we introduce absorbing and critical angles, characterize them, and show that they can be used to give a geometric characterization of absorbing degree two and floating degree three FT configurations.
3.1 Fermat-Torricelli points We call a point x0 a Fermat-Torricelli point (or FT point) of distinct points x1 , . . . , xn in a Minkowski space if x = x0 minimizes x 7→ ∑in=1 k x − xi k. See [60] and [9, Chapter 2] for a discussion of FT points in Euclidean spaces; for Minkowski spaces see [19, 37, 38, 75, 99]; see also [26]. A (star) configuration (of degree n) in a Minkowski space is a set of segments { xxi : i = 1, . . . , n} emanating from the same point x (the center of the configuration), with xi 6= x for all i. A configuration { xxi } is pointed if there is a hyperplane H through x such that the interior of each segment xxi is in the same open half space bounded by H. A configuration { xxi } is spread if it is not contained in any closed
29
3 Fermat-Torricelli points and special angles half space bounded by a hyperplane through x. A floating Fermat-Torricelli configuration (or floating FT configuration) is a configuration { x0 xi : i = 1, . . . , n} such that x0 is an FT point of { x1 , . . . , xn }, and an absorbing Fermat-Torricelli configuration (or absorbing FT configuration) is a configuration { x0 xi : i = 1, . . . , n} such that x0 is an FT point of { x0 , x1 , . . . , xn }. The following well-known proposition states that an FT configuration depends only on the directions of the segments. Although it follows at once from the characterization of FT configurations using subdifferentials (Theorem3.2), it is instructive to note that it can also be proved using elementary Minkowski geometry (i.e., by essentially using only the triangle inequality). Proposition 3.1. In any Minkowski space, 1. if { x0 xi } is a floating FT configuration, then it is also an absorbing FT configuration.
−−→ 2. if { x0 xi } is an FT configuration, then so is { x0 yi } for any yi ∈ x0 xi , yi 6= x0 . Proof. If x = x0 minimizes x 7→ ∑in=1 k x − xi k, then x = x0 also minimizes x 7→ ∑in=0 k x − xi k, since for any x ∈ X we have ∑in=0 k x0 − xi k = ∑in=1 k x0 − xi k ≤ ∑in=1 k x − xi k ≤ ∑in=0 k x − xi k. This shows that floating configurations are also absorbing. Suppose { x0 xi : i = 1, . . . , n} is a floating FT configuration. Without loss of generality x0 = o. Thus x = o minimizes x 7→ ∑in=1 k x − xi k. Let yi 6= o be on − → the ray oxi for each i = 1, . . . , n, say yi = λi xi with λi > 0. Clearly {oxi } is an FT configuration iff {oxi0 } is an FT configuration where xi0 = λxi , for any λ > 0, i.e., we may scale an FT configuration. Thus we may assume without loss of generality that each λi ≤ 1 by making the original FT configuration sufficiently large. Then for any x ∈ X we have n
∑ kyi k
n
=
i =1
∑ (k xi k − kxi − yi k)
i =1 n
≤
∑ (kx − xi k − kxi − yi k)
(since o is an FT point)
i =1 n
≤
∑ (kx − yi k
(by the triangle inequality).
i =1
It follows that o is an FT point of {yi }. The case of an absorbing FT configuration is similar. Durier and Michelot [38] gave the following characterization of FT points, which extends the classical characterization in the case of Euclidean spaces. We give the proof in order to demonstrate the use of the subdifferential calculus. Theorem 3.2 ( [38]). Let x0 , x1 , . . . , xn be points in a Minkowski space. 1. If x0 6= x1 , . . . , xn , then { x0 xi : i = 1, . . . , n} is a floating FT configuration if, and only if, each xi − x0 has a norming functional ϕi such that ∑in=1 ϕi = o.
30
3.1 Fermat-Torricelli points 2. If x0 = x j for some j = 1, . . . , n, then { x0 xi : i = 1, . . . , n, i 6= j} is an absorbing FT configuration if, and only if, each xi − x0 (i 6= j) has a norming functional ϕi such that n
k ∑ ϕi k∗ ≤ 1. i6= j i =1
Proof. Let A = { x0 , . . . , xn }. We use the subdfferential calculus. The point p is an FT point of A if and only if p minimizes the convex function n
f ( x) =
∑ k x − x i k,
i =1
if and only if, n
o ∈ ∂ f ( x ) = ∂ ∑ k x − xi k = i =1
n
∑ ∂ k x − xi k =
i =1
n
∑ ∂ ( x − xi )
i =1
by Lemma 2.10. Sufficiency can also be shown directly as follows: If the purported FT point p 6∈ A then for any x ∈ X, n
n
i =1
i =1 n
n
i =1 n
i =1 n
∑ k xi − p k = ∑ ϕi ( xi − p ) = =
∑ ϕi ( xi − x ) + ∑ ϕi ( x − p )
∑ ϕi (xi − x) + ( ∑ ϕi )(x − p)
i =1 n
=
i =1
∑ ϕi ( xi − x )
i =1 n
≤
∑ k x i − x k,
i =1
while if p = x j ∈ A, then for any x ∈ X,
∑ k xi − p k = ∑ ϕi ( xi − p )
i6= j
i6= j
= ∑ ϕi ( xi − x ) + ∑ ϕi ( x − p ) i6= j
i6= j
= ∑ ϕi ( xi − x) + (∑ ϕi )( x − p) i6= j
i6= j
≤ ∑ k xi − x k + k ∑ ϕi k ∗ k x − p k i6= j
i6= j
n
≤
∑ k x i − x k.
i =1
31
3 Fermat-Torricelli points and special angles Proposition 3.1 follows immediately from the above characterization, as well as the following observation. Corollary 3.3. In any Minkowski space, if {ox1 , . . . , oxn } is a floating FT configuration, then {ox1 , . . . , oxn−1 } is an absorbing FT configuration. It is well-known that there is a unique FT point for any non-collinear set in Euclidean space. The essential property of Euclidean space that ensures uniqueness is its strict convexity. Theorem 3.4. A Minkowski space M is strictly convex if and only if each non-collinear finite subset A ⊂ M has only one FT point. Proof. If X is not strictly convex, then there exist linearly independent x and y such that k x + yk = k xk + kyk. Then x and y are both FT points of {o, x + y}. Conversely, suppose that p and q are distinct FT points of a finite set A. Since the function f : x 7→ ∑ a∈ A k x − ak is convex, its set of minimum points is also convex. Thus each point on the segment pq is an FT point of A. Since A is finite, we may then assume that p, q 6∈ A. By Theorem 3.2, there exist norming functionals ϕ a of a − p, a ∈ A, such that ∑i ϕ a = o. Thus,
∑ k a − pk
=
a∈ A
∑ ϕ a ( a − p) i
=
∑
ϕi ( a − q ) +
a∈ A
≤
∑
ϕ a (q − p)
a∈ A
∑ kaqk = ∑ kapk.
a∈ A
a∈ A
Since there is equality throughout, it follows that each ϕ a is a norming functional also of a − q. Since A is not collinear, we may choose an a such that a, p, q are not collinear. Thus \ a − p and a[ − q are distinct unit vectors with the same norming functional ϕ a . Thus \ a − p a[ − q is a segment on the boundary of the unit ball, hence M is not strictly convex.
3.2 A sufficient condition for an absorbing FT configuration As first noted by Fagnano, the FT point of a set A of 4 non-collinear points in the Euclidean plane can be found very easily. If A is the vertex set of a convex quadrilateral, then the FT point is the point of intersection of the diagonals of the quadrilateral (floating case). Otherwise the FT point coincides with the point of A that lies in the convex hull of the remaining three points (absorbing case). Cieslik [26] generalized this to arbitrary Minkowski planes, in the sense that in the floating case one of the FT points must be as described above, and in the absorbing case this point is the unique FT point. In [60, Proposition 6.4] the absorbing case of Fagnano’s result is generalized to the case where one of the given points lies “in the middle” of the remaining points in a certain sense described in the next theorem, which is the generalizion to Minkowski planes [97]:
32
3.3 Critical and absorbing angles Theorem 3.5 ( [97]). Let p0 , p1 , . . . , pn be distinct points in a Minkowski plane such that for any distinct i, j satisfying 1 ≤ i, j ≤ n the closed angle ^ pi p0 p j contains the reflection in p0 of some pk . Then n is necessarily odd and p0 is an FT point of p0 , . . . , pn . For odd n ≤ 7, and any convex n-gon p1 . . . pn , there always exists p0 such that the hypotheses of the above theorem is satisfied; see [10, 103]. Lemma 3.6. Let n ∈ N be odd and let P be a polygon with vertices
{ x1 , . . . , x n , − x1 , . . . , − x n } in this order. Then n
∑ (−1)i xi =
i =1
1 n (−1)i+1 ( xi+1 − xi ) ∈ P. 2 i∑ =1
Proof. The equation is simple to verify. That the right-hand side is in P follows from (2.2). Note that Lemma 3.6 does not hold for even n. We can now easily prove Theorem 3.5. Proof of Theorem 3.5. By Theorem 3.2 it is sufficient to find norming functionals ϕi −−→ −−→ of pi − p0 such that k∑in=1 ϕi k∗ ≤ 1. We order p1 , . . . , pn such that p0 p1 , . . . , p0 pn are ordered counter-clockwise. If p0 ∈ pi p j for some 1 ≤ i < j ≤ n, we may choose ϕi = − ϕ j . We may therefore assume that p0 ∈ / pi p j for all distinct i, j. Thus for any −−→ i, the open angle ^ pi p0 pi+1 contains a ray opposite some p0 pk . We now show that necessarily n is odd and k ≡ i + (n + 1)/2 (mod n). Since each open angle contains at least one − pk , each open angle contains exactly one such − pk , say − pk(i) . The line through p0 and pk(i) cuts { p1 , . . . , pn } in two open half planes. One half plane contains as many open angles as points pi . Thus n is odd, and k (i ) ≡ i + (n + 1)/2 (mod n). It is now possible to choose norming functionals ϕi of pi − p0 , i ∈ [n], such that ϕ1 , − ϕm+1 , ϕ2 , − ϕm+2 , . . . are consecutive vectors on the unit circle in the dual normed plane. By Lemma 3.6, k∑i ϕi k∗ ≤ 1. A naive generalization of Theorem 3.5 is not possible, even in Euclidean 3-space. For example, using Lemma 3.2 it can be shown that for a regular simplex with vertices xi (i = 1, . . . , 4) there exists a point x5 in the interior of the simplex such that x5 is not a Fermat-Toricelli point of { x1 , . . . , x5 } — we may take any x5 sufficiently near a vertex.
3.3 Critical and absorbing angles We now consider characterizations of degree two absorbing and degree three floating FT configurations in Minkowski planes. In studying such configurations, it is useful to introduce two special types of angles.
33
3 Fermat-Torricelli points and special angles An angle ^x1 x0 x2 in a Minkowski space M is critical if there exists a point x3 6= x0 such that x0 is an FT point of { x1 , x2 , x3 }. Critical angles are a direct generalization of Euclidean 120◦ angles. Note that our term critical angle is at variance with the critical angles of [28] — we have in effect taken Proposition 3 of [28] as our definition of critical angle. −−→ The ray x0 x3 is unique for all critical angles if and only if the Minkowski plane is smooth and strictly convex [36].
3.3.1 General characterizations Lemma 3.7. The following are equivalent in a Minkowski space. 1. ^x1 x0 x2 is a critical angle. 2. There exist norming functionals ϕ1 of x1 − x0 and ϕ2 of x2 − x0 such that k ϕ1 + ϕ2 k = 1. Proof. This is straightforward from the subdifferential calculus. The following characterizations of critical angles are well-known in the literature in the case of smooth, strictly convex planes [19, 36]. However, the generalization to arbitrary planes is simple. Lemma 3.8. The following are equivalent in a Minkowski plane. 1. ^x1 x0 x2 is a critical angle.
−−→ −−→ 2. If a circle (in the norm) with centre x0 intersects the ray x0 x1 at a and the ray x0 x2 at b, then there exist lines ` a , `b , ` supporting the circle, ` a at a and `b at b, such that x0 is the centroid of the triangle formed by ` a , `b , `. It is immediate that in a strictly convex smooth plane, a degree two configuration forming a critical angle can be extended uniquely to a FT configuration of degree three. Related to this is Lemma 2 of [36] which we now prove in a simple way using the dual. Note that the proof in [36] uses Lemma 3.8, while we will use Lemma 3.7. Proposition 3.9 ( [36]). In a strictly convex, differentiable Minkowski plane M, let x1 be any unit vector. Then there exist unique unit vectors { x2 , x3 } such that {ox1 , ox2 , ox3 } is a floating FT configuration. Proof. Let ϕ1 be the unique norming functional of x1 . Since the dual M∗ is strictly convex, the unit circle C1 with center o and unit circle C2 with center − ϕ1 intersect in exactly two points ϕ2 and ϕ3 on opposite sides of the line ` through o and ϕ1 . Obviously, − ϕ1 − ϕ2 is also on both C1 and C2 , and is on the same side of ` as ϕ3 . Therefore, ϕ1 + ϕ2 + ϕ3 = o. Since the original plane is strictly convex, there exist unique unit vectors xi with norming functional ϕi , i = 2, 3. By Lemma 3.7, {ox1 , ox2 , ox3 } is a floating FT configuration. Since this construction is reversable, { x2 , x3 } are unique.
34
3.3 Critical and absorbing angles An angle ^x1 x0 x2 is absorbing if x0 is an FT point of { x0 , x1 , x2 }. Thus { x0 x1 , x0 x2 } is a degree two absorbing FT configuration if and only if ^x1 x0 x2 is absorbing. The following two lemmas furnishes a more direct description of absorbing angles. In particular, an angle is absorbing iff it contains a critical angle. Lemma 3.10. The following are equivalent in a Minkowski space. 1. ^x1 x0 x2 is an absorbing angle. 2. ^x1 x0 x2 contains some critical angle ^x10 x0 x20 . 3. There exist norming functionals ϕ1 of x1 − x0 and ϕ2 of x2 − x0 such that k ϕ1 + ϕ2 k ≤ 1. ∗ ∗ 4. The exposed faces [ x\ 0 − x2 ] of the dual unit ball are at distance at 1 − x0 ] and [ x\ most 1.
Proof. The equivalence between (1) and (3) follows immediately from the subdifferential calculus. The equivalence between (1) and (2) then follows from Lemma 3.7. The equivalence between (3) and (4) is trivial. Corollary 3.11. An angle that contains an absorbing angle is itself absorbing. Lemma 3.12. The following are equivalent in a Minkowski plane. 1. ^x1 x0 x2 is an absorbing angle.
−−→ −−→ 2. If a circle (in the norm) with centre x0 intersects the ray x0 x1 at a and the ray x0 x2 at b, then there exist lines ` a , `b , ` with ` a supporting the circle at a, `b at b, and ` not intersecting the interior of the circle, such that x0 is the centroid of the triangle formed by ` a , `b , `. Proof of Lemmas 3.8 and 3.12. These lemmas can be proved from Lemmas 3.7 and 3.10 as in [19]. We call a Minkowski plane an X-plane if it admits two supplementary absorbing angles (i.e., the union of the two angles is a closed half plane). The class of Xplanes will be very important in our study of the local Steiner problem in Minkowski planes (Chapter 5). Note that Proposition 4 of [28] states (in our terminology) that strictly convex X-planes do not exist. This is disproved in [1] (see Example 6.4). The following theorem, generalizing the characterization of degree three floating FT configurations in the Euclidean plane, will be used inChapter 5. It is surprising that a Euclidean result can be completely generalized to all Minkowski planes. It is again surprising that the proof is not simple. We need the following lemma. Lemma 3.13. In a Minkowski plane, a floating FT configuration of odd degree cannot be pointed.
35
3 Fermat-Torricelli points and special angles Proof. Let the floating FT configuration be { x0 xi : i = 1, . . . , n}, and assume that it is pointed. By Theorem 3.2, each xi − x0 , i ∈ [n], has a norming functional ϕi such that ∑in=1 ϕi = o. Since the configuration is pointed, all ϕi ’s must be contained in the same closed half plane bounded by a line ` through the origin in the dual. Since the sum of the ϕi ’s is o, all the ϕi ’s must lie on `. Therefore, n must be even, with half of the ϕi ’s being equal to some ϕ, and the other half to − ϕ. Theorem 3.14. The configuration {oa1 , oa2 , oa3 } is a floating FT configuration if, and only if, it is not pointed and all angles ^ai oa j are critical. Proof. ⇒ The angles are all critical by definition. By Lemma 3.13, the configuration is not pointed. ⇐ Let Ai be the set of norming functionals of ai (i = 1, 2, 3). Note that if ^ai oa j is a straight angle, then it is critical only if Ai = − A j is a non-degenerate segment. Thus in all cases, A1 , A2 , A3 are not contained in a closed half plane bounded by a line through the origin in the dual plane. We now apply Theorem 3.2 and the following lemma. Lemma 3.15. In a Minkowski plane, let Ai be the intersection of the unit ball with supporting line `i (i = 1, 2, 3) to the unit ball such that for all distinct i, j ∈ {1, 2, 3} there exist ai ∈ Ai and a j ∈ A j such that k ai + a j k = 1 and such that A1 , A2 , A3 are not contained in a closed half plane bounded by a line through the origin. Then there exist ai ∈ Ai , (i = 1, 2, 3) such that a1 + a2 + a3 = o. Proof. Assume without loss of generality that M2 = (R2 , k·k) with `1 is vertical, `2 horizontal lines enclosing the unit square [−1, 1]2 ⊆ R2 . Then the unit ball is contained in [−1, 1]2 . Let A1 = {1} × [−α, β] and A2 = [−γ, δ] × {1}. Then A1 + A2 = [1 − γ, 1 + δ] × [1 − α, 1 + β]. Note that α, γ ≥ 0: If α < 0 or γ < 0, then A1 + A2 would be outside [−1, 1]2 , hence k a1 + a2 k > 1 for all a1 ∈ A1 and a2 ∈ A2 . For similar reasons, A3 must contain a point with x-coordinate ≤ 0, and a point with y-coordinate ≤ 0. Since A1 + A2 intersects the unit ball, there must be a unit vector in [1 − γ, 1] × [1 − α, 1]. Since A1 , A2 , A3 are not in a closed half plane, one of the following cases must occur: 1. − A3 intersects [1 − γ, 1] × [1 − α, 1], in which case we are done. 2. − A3 ⊆ [1 − γ, 1] × [ β, 1 − α) (this is possible only if α + β < 1). 3. − A3 ⊆ [δ, 1 − γ) × [1 − α, 1] (this is possible only if γ + δ < 1). 4. − A3 = A1 . 5. − A3 = A2 . Case 2. A1 + A3 ⊆ {1} × [−α, β] + [−1, −1 + γ] × (−1 + α, β]
= [0, γ] × (−1, 0],
36
3.3 Critical and absorbing angles
Figure 3.1: A pointed FT configuration of even degree in a non-strictly convex plane which does not contain a unit vector, a contradiction. Case 3. A similar contradiction is obtained. Case 4. A1 + A3 = {0} × [−α − β, α + β]. Since A1 + A3 must contain a unit vector, 1 ∈ [− −1 + α − β, α + β], i.e. α + β1≥ 1. Therefore −β ≤ − α < α, and the −1 1 0 vector −1+α ∈ A3 . Then vectors −α ∈ A1 , 1 ∈ A2 , −1+α ∈ A3 have sum o. Case 5. Similar to Case 4. We remark that Lemma 3.13 is part of Corollary 4.1 in [75]. Note that there exist pointed floating FT configurations of any even degree in any Minkowski plane that is not strictly convex (Figure 3.1). (In the last part of Corollary 4.1 of [75] it is incorrectly stated that absorbing FT configurations of even degree cannot be pointed—in fact an absorbing angle is an absorbing FT configuration of degree 2.) Any finite set of points in a Minkowsi plane has a FT point in the convex hull of the set [106], and when the number of points is odd, then all FT points are contained in the convex hull [75, Corollary 4.1]. Below we show that in the λ-geometry with λ 6≡ 2(mod 3) there exist star configurations of degree three with all angles absorbing, without the configuration being floating FT. Thus the requirement in Theorem 3.14 that the angles be critical cannot be weakened to absorbing.
3.3.2 The λ-geometry Recall that the unit ball of the λ-geometry is an affine regular 2λ-gon. Note the following easily proved facts. Lemma 3.16. 1. The dual of the λ-geometry is again the λ-geometry. Let the vertices of the dual unit ball be ϕ1 , . . . , ϕ2λ , in this order, with subscripts considered to be modulo 2λ throughout. 2. For any i and any k ∈ {0, 1, . . . , λ}, ∗
k ϕi − ϕi +k k =
2 sin πk 2λ π 2 sin πk 2λ / cos 2λ
if if
λ ≡ k (mod 2), λ 6≡ k (mod 2),
37
3 Fermat-Torricelli points and special angles λ ≡ 0 (mod 3)
λ ≡ 1 (mod 3)
λ ≡ 2 (mod 3)
2λ −1 3
2λ−2 3
sides
sides
2λ −1 3
2λ−1 3
sides
2λ−1 3
sides
sides
2λ−2 3
sides
2λ −1 3
2λ−2 +1 3
2λ−1 3
sides
sides
sides
Figure 3.2: Degree 3 floating FT configurations in the λ-geometry 3. and
=1 1
if if if
k = λ/3, k < λ/3, k > λ/3.
4. In the λ-geometry, ^xoy is absorbing if, and only if, it contains at least d 2λ 3 e−1 consecutive full sides of the unit circle, and ^xoy is critical if, and only if, it is furthermore covered by some b 2λ 3 c + 1 consecutive sides of the unit circle. Proof. (1) is obvious by symmetry considerations. (2) and (3) follow from elementary trigonometry. (4) follows from (3) and Lemmas 3.7 and 3.10. Note that in a similar (but more laborious) way the absorbing and critical angles may be described in any normed plane with polygonal unit ball. We now apply Theorem 3.14 to the above description to describe all floating FT configurations of degree 3. By Lemma 3.16 each angle of a degree 3 floating FT configuration centered at o must contain at least d 2λ 3 e − 1 full edges of the unit 2λ circle, and be covered by b 3 c + 1 full edges. Also, the configuration must be nonpointed. We then obtain the following description. Proposition 3.17. A degree 3 floating FT configuration in the λ-geometry must be as in Figure 3.2. Note that if λ 6≡ 2(mod 3) then there exist non-Steiner configurations of degree 3 with all angles absorbing. This shows that we cannot replace critical angles by absorbing angles in Theorem 3.14.
38
Chapter
4
The local Steiner problem The Steiner problem is the problem of finding an SMT if it exists (or all SMTs) for a given set of points in a metric space. It seems that Gauss, in a letter to Schumacher, first considered what would today be considered to be a Steiner minimal tree [26]. ¨ The problem was independently stated by Jarn´ık and Kossler in 1934 [56], the (inaccurate) name of the problem originated in [31], and the term Steiner minimal tree ¨ in [44]. Around the same time as Jarn´ık and Kossler, Menger [78] and Mimura [79] also considered Steiner minimal trees in their study of definitions of curve length. The Steiner problem in the rectilinear (taxicab) plane and in the λ-geometry is important in certain aspects of VLSI design [58,91]. The Steiner problem has first been considered in the rectilinear plane by Hanan [50], and in general normed planes by Cockayne [28]. See [26, 54, 55] for monographs on the Steiner problem. There is a vast literature on Steiner minimal trees, mostly in graphs, Hamming space and related word spaces, the Euclidean plane, and the Manhattan plane, with applications in VLSI design [58] and phylogenetics [27]. Cockayne [28] considered Minkowski planes (two-dimensional normed spaces). Cieslik initiated the study of SMTs in general finite-dimensional normed spaces [21, 23]. Recently, other norms besides Euclidean and Manhattan have found applications [20, 64, 65, 67–69], while the geometry of certain high-dimensional normed spaces is related to the metric spaces found in the mathematical study of phylogenetic trees [27]. These trees are also of interest in differential geometry; see especially the work of Morgan and his students [1, 30, 63, 81, 82].
4.1 Steiner minimal trees Although we already gave an intuitive explanation of Steiner minimal trees in Chapter 1, we here give a formal definition. Let N ⊂ Md be a finite, non-empty set of points in a Minkowski space. A spanning tree T of N is an acyclic connected graph with vertex set N. Denote its edge set by E( T ). A Steiner tree T of N is a spanning tree of some finite V ⊂ Md such that N ⊆ V and such that the degree of each vertex in V \ N is at least 3. The vertices in N are the terminals of the Steiner tree T, and the vertices in V \ N the Steiner points of T. The length of a tree T in Md
39
4 The local Steiner problem is
`( T ) :=
∑
k x − y k.
xy∈ E( T )
A Steiner minimal tree (SMT) of S is a Steiner tree of S of smallest length. The requirement that Steiner points have degree at least 3 is for technical convenience, since Steiner points of degree at most 2 can easily be eliminated using the triangle inequality without making the tree longer. It is easily seen that the number of Steiner points is at most |S| − 2. It then follows by a simple compactness argument that any non-empty finite S has a SMT [28]. In the literature, adjacent vertices of Steiner trees in the rectilinear plane are conventionally connected by a shortest path consisting of a horizontal segment and a vertical segment, which is dictated by practical applications. There is a similar convention with the λ-geometry. However, in our definition of SMTs we assume that the edges of a Steiner tree in a normed space are realized by straight line segments between adjacent vertices. This is no loss in generality, since any segment xy in a normed plane with polygonal unit ball may be replaced by a broken segment xz ∪ zy, with x[ − z and y[ − z extreme points of the unit ball.
4.2 Formulation of the local Steiner problem Conceptually, there are two aspects to the Steiner problem: the first is searching through all candidate Steiner trees, the second is finding the optimal Steiner points in a space, given a candidate tree. The first aspect is a problem in combinatorial optimization, and it is not surprising that NP-hardness raises its head there (see [42,43]). A large part of the literature therefore concentrates on (exponential time) algorithms for the Euclidean and rectilinear planes with reasonable running times for reasonably small sets (see e.g. [40] and [107]). The second aspect is geometrical, and finitely solvable in e.g. the cases of Euclidean [77] and rectilinear planes [50]. However, the properties of Steiner points in these two planes are very different. In particular, in the Euclidean plane Steiner points must have degree 3, while in the rectilinear plane they can have degree 4. An attempt at bridging the gap between the two planes is made by the λ-geometry, where the rectilinear plane is the 2-geometry, and the Euclidean plane is approximated as λ → ∞. It would be desirable to unify the various descriptions of the local properties of vertices in SMTs in different normed planes. This is the purpose of this paper. In particular, we consider what we call the local Steiner problem, that of describing the possible geometric structures of the edges emanating from a terminal or Steiner point in an SMT. We now formalize this. A star is an SMT such that one of the vertices, called the center of the star, (either a Steiner point or a terminal) is joined to all the other vertices. The vertex figure of a vertex in an SMT is the set of vectors from the center of the star to the other vertices. A configuration { xxi : i ∈ [n]} is a Steiner configuration [terminal configuration] if it is part of some SMT with x as Steiner point [terminal, respectively]. Equivalently,
40
4.3 Degree quantities
{ xxi : i ∈ [n]} is a Steiner configuration iff it is an SMT of { xi : i ∈ [n]}, and it is a terminal configuration iff it is an SMT of { x, x1 , . . . , xn }. We now formally state the The Local Steiner Problem. Given a metric space X, describe all Steiner and terminal configurations in X. We collect some basic properties of these configurations, analogous to Proposition 3.1. Proposition 4.1. In any Minkowski space, 1. a Steiner configuration is a floating FT configuration, 2. a terminal configuration is an absorbing FT configuration, 3. a Steiner configuration is also a terminal configuration,
−−→ 4. if { x0 xi } is a Steiner [terminal] configuration, then so is { x0 yi } for any yi ∈ x0 xi , yi 6= x0 , 5. All angles in a Steiner or terminal configuration are absorbing. Proof. In (1) and (2), if the configuration were not FT, then the SMT could be shortened. Statements (3) and (4) are proved analogously to Proposition 3.1. If some ^xi xx j is not absorbing, the two edges xxi and xx j can be replaced by a shorter subtree, which proves (5). Surprisingly, the converse of Proposition 4.1(5) is true for terminal configurations in a Minkowski plane, but not for Steiner configurations, where extra conditions are needed to achieve a converse (see Theorem 5.1 in Chapter 5). Proposition 4.2. In a Minkowski plane, a Steiner configuration is never pointed. Proof. Draw a picture and use the triangle inequality. A terminal configuration can be pointed; see Section 6.6. If T is an SMT of N, then T is clearly still an SMT of any N 0 where N ⊆ N 0 ⊆ V ( T ). Therefore, if x ∈ N has neighbors x1 , . . . , xk ∈ V in some SMT, then the star joining x to each xi , i ∈ [k ], is a SMT of { x, x1 , . . . , xk }. Thus, to characterize the neighborhoods of terminals in SMTs, it is sufficient to characterize SMTs which are stars with the center a terminal.
4.3 Degree quantities We denote the maximum degree of a terminal in an SMT in Md , with the maximum taken over all possible SMTs, by τ (Md ). Also, let σ(Md ) be the maximum degree of a Steiner point in an SMT in Md , where the maximum is taken over all possible SMTs in Md . t follows from Proposition 4.1 that σ(Md ) ≤ τ (Md ). Cieslik [21] proved that τ (Md ) is bounded above by the Hadwiger number or translative kissing number of the unit ball B(Md ), i.e., the maximum number of mutually non-overlapping translates of B(Md ) that all touch B(Md ). He proved this
41
4 The local Steiner problem
Figure 4.1 more generally for minimal spanning trees. Cieslik also made the following conjecture: Conjecture 4.3 (Cieslik [21]). The maximum degree of an SMT in any d-dimensional Minkowski space Md satisfies τ (Md ) ≤ 2d+1 − 2, with equality if and only if Md is isometric to MdZ . See Section 2.5 for the definition of MdZ . Since any edge joining two points in an SMT can be replaced by a piecewise linear path consisting of segments parallel to the vectors pointing to the extreme points of the unit ball, we obtain the following well-known lemma (essentially in [35]). Lemma 4.4. If the unit ball of Md is a polytope with v vertices, then τ (Md ) ≤ v. It follows that indeed τ (MdZ ) ≤ 2d+1 − 2. Cieslik [23] proved the case d = 2 of his conjecture. However, in Chapter 8 we will find the correct value of τ (MdZ ), which turns out to be less than 2d+1 − 2 for all d ≥ 3 (see below), thus partially disproving the conjecture. It is in fact easy to see that τ (M3Z ) < 14 by shortening the star connecting the origin to the vertices of the rhombic dodecahedron. If there existed a SMT with a given point of degree 14, then the star joining o to the vertices of the unit ball Z3 would have to be a SMT of these 14 vertices together with the origin. In Fig. 4.1 on the left we show three of the fourteen edges from the origin to the vertices. The length of this tree is 3. On the right we show that the tree can be shortened by introducing two Steiner points. This new tree has length 5/2. From the fact that this subconfiguration is forbidden, it is not difficult to see that τ (M3Z ) ≤ 10. This turns out to be the correct value (Figure 4.2). d +2 Theorem 8.1. For any d ≥ 3, τ (MdZ ) = (b(d+ 2)/2c).
The proof is in Chapter 8, where we also describe all possible vertex figures of a terminal in an SMT in MdZ . Corollary 8.6. In the Minkowski space MdZ , the star joining o to any set A of non-zero points is an SMT of A ∪ {o} if and only the star joining o to any subset B of A of size at most 4 is an SMT of B ∪ {o}.
42
4.3 Degree quantities For all d ≥ 2 there is, up to isometries, a unique configuration attaining the maximum degree (Theorem 8.7). These spaces give the largest known degrees of SMTs in Minkowski spaces of dimensions 2 to 6, in fact τ (M3Z ) = 10, τ (M4Z ) = 20, τ (M5Z ) = 35, and τ (M6Z ) = 70. For d ≥ 7, τ (`d∞ ) = 2d is larger. It is not at all clear whether MdZ maximizes τ (Md ). Morgan [81, p. 42], [82, p. 129] asks the following question: Conjecture 4.5 (Morgan [81, 82]). The maximum degree of a Steiner point in an SMT in any d-dimensional Minkowski space Md satisfies σ(Md ) ≤ 2d . The asymptotically best known upper bound for both conjectures is σ (Md ) ≤ τ (Md ) ≤ c2d d2 log d, as will be shown in Chapter 9. In Chapter 5 we show that σ(M2 ) ≤ 4 for all Minkowski planes. There are many two-dimensional spaces attaining σ(M2 ) = 4, and they will all be characterized in Chapter 5. In Chapter 8 we also give an upper and lower bound for σ(MdZ ) that is asymptotically correct up to a factor of 2. √ d +1 d d Theorem 8.8. For any d ≥ 3, σ(MdZ ) ≥ (b(d+ 1)/2c). Therefore, σ (MZ ) = Θ (2 / d ). The sharp upper bound for spaces Md with differentiable norms is σ(Md ) ≤ τ (Md ) ≤ d + 1 [63, 96]. It is well-known that σ(`d∞ ) = τ (`d∞ ) = 2d [81] and σ(`1d ) = τ (`1d ) = 2n (see also Section 8.2). For the ` p norm, 1 < p < ∞, we have 3 ≤ σ(`dp ) ≤ p τ (`dp ) ≤ 7 if p > 2, n ≥ 2, and min{n, ( p−1) ln 2 } ≤ σ(`dp ) ≤ τ (`dp ) ≤ 2 p/( p−1) if 1 < p < 2, n ≥ 3; see [96], where more detailed estimates are obtained. Conger [30] showed that σ(R3 , k·k1 + λk·k2 ) ≥ 6 for all 0 < λ ≤ 1. These norms 2 are piecewise √ C ∞ and elliptic. In [1] it is √shown that σ(R , k·k1 + λk·k2 ) = 4 for all 0 < λ ≤ 2 + 2. The value λ = 2 + 2 is sharp as demonstrated in Section 6.4. More generally, we show the following in Chapter 8. Theorem 8.12. Let λ√> 0, √ and let Md = (Rd , k·k1 + λk·k2 ). If λ ≤ 1 then σ (Md ) ≤ τ (Md ) ≤ 2d. If λ ≤ d/( d − 1) then τ (Md ) ≥ σ(Md ) ≥ 2d.
Figure 4.2
43
4 The local Steiner problem In this regard Conger made the following conjecture [82, p. 128]. Conjecture 4.6 (Conger [82, p. 128]). For any d-dimensional normed space Md with piecewise differentiable, elliptic norm, σ(Md ) ≤ 2d. Our results are based on a characterization of the vertex figures of terminals and of Steiner points in SMTs in arbitrary Minkowski spaces (Chapter 7). This characterization is found using the subdifferential calculus (Chapter 2). The characterization of vertex figures of terminals enables us to reduce the determination of τ (MdZ ) to a purely combinatorial problem in the extremal theory of finite sets. The characterization of vertex figures of Steiner points in the case of MdZ is not so easily reducible to combinatorics, hence the partial results of Theorem 8.8. This situation can be compared to Theorem 5.1, where the characterization of terminals in an arbitrary two-dimensional Minkowski space is much simpler than the characterization of Steiner points. However, the norm k·k1 + λk·k2 is sufficiently simple so that the maximum degree of a Steiner point can be determined.
4.4 Other known results Hanan [50] mentions that the maximum possible degree of a Steiner or terminal in the rectilinear plane is four. Alfaro et al. [1] shows that there even exists a normed plane with a strictly convex and piecewise smooth unit circle admitting degree four Steiner points. They also show that the degree of a Steiner point is at most four in all elliptic planes (i.e. the unit circle is C2 and has positive inward curvature everywhere). Upon examination their proof in fact holds for all strictly convex planes and also shows that the degree of a terminal is at most four. They furthermore show that in a strictly convex plane, alternate edges of a any degree four Steiner point must be collinear. For smooth and elliptic normed planes they show that the maximum degree of a Steiner point is three. Lawlor and Morgan [63] generalize this last fact to all differentiable norms. Cieslik [22] shows that the degree of a vertex is at most five if the plane is not the 3-geometry (where the maximum degree of a terminal is 6), and also that the degree of a vertex is at most four if the plane is either smooth or strictly convex. He also shows that for any finite point set in a normed plane there exists an SMT with all vertices of degree at most four.
44
Chapter
5
Solving the local Steiner problem in two dimensions In this chapter we solve the local Steiner problem for all Minkowski planes. Our approach is geometrical and unifies various previous results known for specific norms. The absorbing and critical angles introduced in Chapter 3 stand in the center of this analysis. As corollaries we show that the maximum possible degree of a Steiner point and of a terminal are equal, and equal 3 or 4, except if the unit ball is an affine regular hexagon, where the maximum degree of a Steiner point is 4, and of a terminal is 6. We also characterize the planes where the maximum degree is 4, the so-called X-planes, and present examples. In particular, in the λ-geometry (where the unit ball is a regular polygon with 2λ sides), Steiner points of degree 4 exist if and only if λ = 2, 3, 4 or 6.
5.1 Characterization of planar terminal and Steiner configurations Theorem 5.1. Consider any normed plane M2 . 1. A star configuration in M2 is terminal if and only if all angles determined by the configuration are absorbing. 2. Degree four terminal configurations exist in M2 if and only if M2 is an X-plane. All degree four terminal configurations are spread with the single exception of the configuration in the 3-geometry in Figure 5.1. 3. Terminal configurations of degree five or six exist in M2 if and only if the plane is isometric to the 3-geometry, and must be as in Figure 5.2. 4. Terminal configurations of degree more than 6 do not exist in M2 . 5. A star configuration of degree 3 in M2 is Steiner if and only if the configuration is floating FT, if and only if the configuration is not pointed and all angles are critical.
45
5 Solving the local Steiner problem in two dimensions
Figure 5.1: An exceptional degree four terminal configuration
Figure 5.2: Regular configurations of degree five and six exist only in the 3geometry 6. A star configuration of degree 4 in M2 is Steiner if and only if all angles are absorbing and some two opposite edges are collinear, i.e. two angles are supplementary. 7. Steiner configurations of degree 4 exist in M2 if and only if the plane is an X-plane. 8. Steiner configurations of degree more than 4 do not exist in M2 . We state a few noteworthy corollaries of the Main Theorem. See Chapter 6 for further applications. Corollary 5.2. The maximum degrees of Steiner and terminals in any normed plane are as in Table 5.1. The next corollary improves an example of [1]. Corollary 5.3. There exists a strictly convex X-plane (hence degree four Steiner points exist), with the unit ball smooth at each boundary point except one pair of antipodal points, where degree four Steiner configurations exist (Figure 5.3). Proof. See Section 6.5 in the next chapter. Plane M2 Non-X-plane X-planes except M2Z M2Z
σ(M2 ) 3 4 4
τ ( M2 ) 3 4 6
Table 5.1: Maximum vertex degree in SMTs for all Minkowski planes
46
5.2 Proof of the characterization
Figure 5.3: An almost smooth X-plane Corollary 5.4. The λ-geometry is an X-plane if, and only if, λ ∈ {2, 3, 4, 6}. Proof. See Section 6.7 in the next chapter.
5.2 Proof of the characterization We subdivide the proof into a sequence of propositions. Proposition 5.5. A degree 2 configuration is terminal if, and only if, the configuration is an absorbing FT configuration. A degree 3 configuration is Steiner if, and only if, the configuration is a floating FT configuration. Proof. In both cases there are three terminals x1 , x2 , x3 . Any Steiner tree connecting these three points is a tree connecting x1 , x2 , x3 to some point which is either a Steiner point or one of the xi ’s. It follows that an SMT of { x1 , x2 , x3 } is a FT configuration of { x1 , x2 , x3 } and conversely. Lemma 5.6. Let e1 , e2 , e3 be unit vectors in a Minkowski plane such that the origin o ∈ conv{e1 , e2 , e3 }. Then ke1 + e2 + e3 k ≤ 1. Proof. Let o=
3
3
i =1
i =1
∑ αi ei , ∑ αi = 1,
Then
αi ≥ 0,
i = 1, 2, 3.
3
e1 + e2 + e3 =
∑ (1 − 2αi )ei ,
i =1
hence
3
k e1 + e2 + e3 k ≤
∑ |1 − 2αi |.
i =1
Also, for each i = 1, 2, 3, −αei = ∑ j6=i α j e j , αi = k ∑ α j e j k ≤ 1 − αi , j 6 =i
and 1 − 2αi ≥ 0. It follows that ke1 + e2 + e3 k ≤ ∑3i=1 (1 − 2αi ) = 1.
47
5 Solving the local Steiner problem in two dimensions Lemma 5.7. In a Minkowski plane, let Ai be the intersection of the unit ball with a tangent line `i (i = 1, 2, 3), such that for any distinct i, j ∈ {1, 2, 3}, there exist ai ∈ Ai and a j ∈ A j such that k ai + a j k ≤ 1. Then there exist ai ∈ Ai , (i = 1, 2, 3) such that k a1 + a2 k ≤ 1 and k a1 + a2 + a3 k ≤ 1. Proof. As in the proof of Lemma 3.15, we have without loss of generality that the unit ball is contained in [1−, 1]2 ,and A1 = {1} × [−α, β], A2 = [−γ, δ] × {1} for some α, β, γ, δ ≥ 0. Let a1 = −1α , a2 = −1γ . Then k a1 + a2 k ≤ 1. We now show that a3 ∈ A3 can be chosen arbitrarily. Since A3 must contain a point with x-coordinate ≤ 0 and a point with y-coordinate ≤ 0, one of the following cases must occur. 1. A3 intersects [−1, γ] × [−1, α] 2. A3 ⊆ [δ, 1] × [ β, 1] In Case 1 we have for any a3 ∈ A3 that o ∈ conv{ a1 , a2 , a3 }, hence k a1 + a2 + a3 k ≤ 1 by Lemma 5.6. In Case 2, since A3 must have a point with x-coordinate ≤ 0 and a point with y-coordinate ≤ 0, we obtain β = δ = 0 and A3 = e1 e2 , where e1 = 10 , e2 = 01 . Then k a1 + a2 k = (−γ + 1) + (1 − α) ≤ 1, hence α + γ ≥ 1. Similarly, from k a1 + e2 k ≤ 1 it follows that α ≥ 1. Thus, α = 1. Also, from k a2 + e1 k ≤ 1 it follows that γ = 1. Thus a1 + a2 = o, and we can choose any a3 ∈ A3 . Proposition 5.8. If the three angles in the configuration {ox1 , ox2 , ox3 } are all absorbing, then the configuration is terminal. Proof. We have to show that {ox1 , ox2 , ox3 } is an SMT of {o, x1 , x2 , x3 }. Up to a permutation of { x1 , x2 , x3 }, any Steiner tree T connecting the terminals o, x1 , x2 , x3 has edge set {os1 , x1 s1 , s1 s2 , x2 s2 , x3 s2 } for some points s1 , s2 that are either Steiner points or coincide with the terminals o, x1 , x2 , x3 (if e.g. s1 = x1 then we discard x1 s1 from E( T )). Since all angles ^xi ox j are absorbing, we can use Lemma 5.7 to find norming functionals ϕi of xi (i = 1, 2, 3) such that k ϕ2 + ϕ3 k∗ ≤ 1 and k ϕ1 + ϕ2 + ϕ3 k∗ ≤ 1. Then 3
∑ koxi k = ϕ1 (x1 ) + ϕ2 (x2 ) + ϕ3 (x3 )
i =1
= ϕ1 ( x1 − s1 ) + ϕ2 ( x2 − s2 ) + ϕ3 ( x3 − s2 ) + ( ϕ2 + ϕ3 )(s2 − s1 ) + ( ϕ1 + ϕ2 + ϕ3 )(s1 ) ≤ k x1 − s1 k + k x2 − s2 k + k x3 − s2 k + k s2 − s1 k + k s1 k = `( T )
Proposition 5.9. A configuration {oxi : i = 1, 2, 3, 4} is Steiner if, and only if, the four angles are two pairs of supplementary absorbing angles.
48
5.2 Proof of the characterization x2 a x1
d e
b x3
o
c x4 Figure 5.4
Proof. =⇒ Let the configuration be Steiner. By Proposition 4.1 all angles are absorbing. Suppose that no two angles are supplementary. We now show that the configuration must be spread. −→ −→ By Proposition 4.2 it is not pointed. Suppose ox1 and ox4 are opposite, with ox2 and ox3 on one side of the line through x1 and x4 . We then reroute around o as follows. Choose ai ∈ oxi (i = 1, 2, 3, 4) such that a1 a2 k ox3 , a2 a3 k ox1 and a3 a4 k ox2 . Then the tree with edges
{ x1 a1 , x2 a2 , x3 a3 , x4 a4 , a1 a2 , a2 a3 , a3 a4 } is shorter, a contradiction. Therefore, the configuration is spread, and we may assume without loss of generality that − x1 ∈ int ^x3 ox4 and − x4 ∈ int ^x1 ox2 . Choose a ∈ ox2 , b ∈ ox3 , c ∈ ox4 , d ∈ oa, e ∈ ox1 such that oabc and obde are parallelograms. (Since − x4 ∈ int ^x1 ox2 , d will be on oa.) See Figure 5.4. Since ^aob is absorbing, k abk ≥ koak, hence kock ≥ kbck. It follows that the rerouted tree { x3 b, x4 b, bd, x2 d, x1 d} is shorter by at least kodk, a contradiction. −→ −→ ⇐= Assume without loss of generality that ox1 and ox3 are opposite. We have to show that {ox1 , . . . , ox4 } is an SMT of { x1 , x2 , x3 , x4 }. Any Steiner tree T connecting { x1 , . . . , x4 } must be, up to permutation of { x2 , x4 }, either TA or TB in Figure 5.5 for some points s1 , s2 each of which is either a Steiner point or coincides with one of x1 , . . . , x4 . Applying Lemma 6.2 to absorbing angles ^x1 ox2 , ^x2 ox3 and ^x3 ox4 , we find that the unit ball has a common tangent parallel to x1 x3 at boundary points xb2 and − xb4 . Let ϕ2 be the corresponding norming functional of x2 and − x4 . We now treat the two cases in turn.
49
5 Solving the local Steiner problem in two dimensions TA
x1 s1 x3
TB
x1
x2
s1
s2 x4
x4
x2 s2 x3
Figure 5.5 TA : Choose any norming functional ϕ1 of x1 . Then 4
∑ koxi k = ϕ1 (x1 ) + ϕ2 (x2 ) − ϕ1 (x3 ) − ϕ2 (x4 )
i =1
= ϕ1 ( x1 − s1 ) − ϕ1 ( x3 − s1 ) + ϕ2 ( x2 − s2 ) − ϕ2 ( x4 − s2 ) ≤ k x1 − s1 k + k x3 − s1 k + k x2 − s2 k + k x4 − s2 k ≤ `( TA ).
(5.1)
TB : Assume without loss of generality that − x4 ∈ ^x1 ox2 . Since ^x2 ox3 is absorbing, there is a norming functional − ϕ1 of x3 and ϕ20 of x2 such that k− ϕ1 + ϕ20 k∗ ≤ 1. Since ^x2 ox3 is contained in the absorbing angle ^x3 o(− x4 ), we must have k− ϕ1 + ϕ2 k∗ ≤ 1. Then 4
∑ koxi k = ϕ1 (x1 ) + ϕ2 (x2 ) − ϕ1 (x3 ) − ϕ2 (x4 )
i =1
= ϕ1 ( x1 − s1 ) − ϕ2 ( x4 − s1 ) + ( ϕ1 − ϕ2 )(s1 − s2 ) − ϕ1 ( x3 − s2 ) + ϕ2 ( x2 − s2 ) (5.2) ≤ k x1 − s1 k + k x4 − s1 k + k s1 − s2 k + k x2 − s2 k + k x3 − s2 k ≤ `( TB ). Proposition 5.10. If the four angles of a star configuration {ox1 : i = 1, 2, 3, 4} are all absorbing, then the configuration is terminal. The configuration is furthermore spread, except if the plane is isometric to the 3-geometry, and then the configuration is as in Figure 5.1. Proof. By Proposition 5.9 it is sufficient to consider the case where no two angles are supplementary (since Steiner configurations are terminal). There are three cases to consider. Case 1 The configuration is pointed. Case 2 The configuration is contained in a closed half plane bounded by a line containing two edges of the configuration. Case 3 The configuration is spread. Case 1. Lemma 6.2 easily gives a contradiction.
50
5.2 Proof of the characterization x2 p
r
a x1
x3 q
b
o
x4 Figure 5.6
In the remaining cases we now find unit functionals ϕ1 , ϕ2 such that ϕ1 is a norming functional of x1 and − x3 , ϕ2 a norming functional of x2 and − x4 , and ϕ2 − ϕ1 of x2 and x3 . −→ −→ Case 2. Let ox1 and ox4 be opposite. It follows from Lemma 6.2 that the unit ball equals conv{± xbi : i = 1, 2, 3, 4}, and xb1 xb2 parallel to ob x3 , xb2 xb3 parallel to ob x1 , and xb3 xb4 parallel to ob x2 . Thus the unit ball is an affine regular hexagon, and the plane is isometric to the 3-geometry. By considering the dual, the required unit functionals ϕ1 and ϕ2 are easily found. Case 3. As in the proof of Proposition 5.9, we may assume without loss of generality that − x1 ∈ int ^x3 ox4 and − x4 ∈ int ^x1 ox2 , and use Lemma 6.2 to find that xb3 (− xb1 ) is a segment contained in the unit circle parallel to ox2 , and xb2 (− xb4 ) is a segment contained in the unit circle parallel to ox3 . Let ϕ1 be the corresponding norming functional of x1 and − x3 , and ϕ2 of x2 and − x4 . As in Case 2, we now show that ϕ2 − ϕ1 is a norming functional of x2 and x3 . Choose a, p ∈ ox2 , b, q ∈ ox3 , r ∈ ^x2 ox3 such that pr is parallel to ox4 , qr is parallel to ox1 , ar is parallel to ox3 , and br is parallel to ox2 (Figure 5.6). Then k p − r k = k p − ak and kq − r k = kq − bk. Also, since ^x2 ox3 is absorbing, kqk + k pk ≤ k p − r k + kq − r k + kr k. It follows that k ak + kbk = kr k. Hence, xb2 xb3 is a segment on the unit circle. Since xb3 (− xb1 ) is parallel to ox2 , we have ϕ1 ( x2 ). Similarly, ϕ2 ( x1 ) = 0. Hence ( ϕ2 − ϕ1 )( x2 ) = ( ϕ2 − ϕ1 )( x3 ) = 1, and it follows that ϕ2 − ϕ1 is a norming functional of x2 and x3 . We now complete the proof in a similar way as the proof of Proposition 5.9. Since ϕ1 , ϕ2 , ϕ2 − ϕ1 are the only functionals we will use, we may reduce cases by simultaneously interchanging x2 ↔ x3 and x1 ↔ x4 . Then there are nine possible Steiner trees connecting {o, x1 , x2 , x3 , x4 }, as in Figure 5.7. (As before, s1 , s2 , s3 are either Steiner points or coincide with terminals.)
51
5 Solving the local Steiner problem in two dimensions
o
T1
x2
T2
s1
s3 s1
s3
s2
x1
x3 x2
x1
s2 x4
x4
x1
x1
x2
x2
s3
s2 x2 x3
s2 s3
s2
o
x3 x1
x3 x3
T9 s1
x1
s1
o
x4
T8 s3
x2
Figure 5.7
52
x4 s3
s2
o
o
x4
x4
s1 x2
s1
x2
T6
s3
s2
x1
T7
x3
x3
s3
x3
o
x3
T5
s1
s3
s2 x4
x4
s1
s2
o
o
T4
x1
T3
s1
o
x4 x2
x1
5.2 Proof of the characterization With T1 we perform the following calculation: 4
∑ k xi k
= ϕ1 ( x1 ) + ( ϕ2 − ϕ1 )( x2 ) + ( ϕ2 − ϕ1 )( x3 ) − ϕ2 ( x4 )
i =1
= ϕ1 ( x1 − s1 ) + ( ϕ2 − ϕ1 )( x2 − s1 ) + ( ϕ2 − ϕ1 )( x3 − s2 ) − ϕ2 ( x4 − s2 ) + ϕ2 (s1 − s3 ) − ϕ1 (s2 − s3 ) + ( ϕ2 − ϕ1 )(s3 ) ≤ `( T1 ). With T2 we perform the following calculation: 4
∑ k xi k
= ϕ1 ( x1 ) + ϕ2 ( x2 ) + ( ϕ2 − ϕ1 )( x3 ) − ϕ2 ( x4 )
i =1
= ϕ1 ( x1 − s3 ) + ϕ2 ( x2 − s1 ) + ( ϕ2 − ϕ1 )( x3 − s2 ) − ϕ2 ( x4 − s2 ) + ϕ2 (s3 − o) + ϕ1 (s1 − s2 ) + ( ϕ2 − ϕ1 )(s3 − s1 ) ≤ `( T2 ). Tree T3 is at least as long as the forest {ox2 , x1 s, x3 s, x4 s}. However,
k x1 k + k x3 k + k x4 k = ϕ1 ( x1 ) + ( ϕ2 − ϕ1 )( x3 ) − ϕ2 ( x4 ) = ϕ1 ( x1 − s) + ( ϕ2 − ϕ1 )( x3 − s) − ϕ2 ( x4 − s) ≤ k x1 − s k + k x3 − s k + k x4 − s k. Trees T4 , T5 , T6 are easily seen to be at least as long as TA in Figure 5.5. Calculation (5.1) shows that this tree has length at least ∑4i=1 k xi k. Trees T7 , T8 , T9 are easily seen to be at least as long as TB in Figure 5.5. Then calculation (5.2) is used. Proposition 5.11. No SMT in a normed plane has Steiner points of degree 5 or 6. The only normed planes where terminal configurations of degree at least 5 exist are planes isometric to the 3-geometry, where the terminal configurations must be as in Figure 5.2. Proof. Let o be a degree 5 vertex in a SMT with neighbours x1 , . . . , x5 . Then all angles ^xi ox j are absorbing. Let Ai be the set of norming functionals of xi for each i = 1, . . . , 5. Then each Ai is a segment or a singleton on the dual unit circle. In the dual normed plane, choose among all triangles 4oϕψ, where ϕ and ψ are on the dual unit circle, one 4oϕ1 ϕ2 of maximum area. Then the dual unit circle is contained in conv{± ϕ1 ± ϕ2 }. It is possible to choose ϕ1 and ϕ2 such that none of {± ϕ1 , ± ϕ2 } is contained in the interior of any Ai that is a segment. It follows that some two Ai ’s, say A1 and A2 , are contained in one of the four quadrants ^(± ϕ1 )o (± ϕ2 ). Since there exist ψ1 ∈ A1 and ψ2 ∈ A2 such that kψ1 + ψ2 k∗ ≤ 1, it follows that ϕ1 + ϕ2 is a unit vector and Ai is the segment joining ϕi to ϕ1 + ϕ2 , (i = 1, 2). By maximality of 4oϕ1 ϕ2 , it follows that the dual unit ball is conv{± ϕ1 , ±( ϕ1 + ϕ2 )}. Thus we have that the dual plane is isometric to the 3-geometry. Therefore, the original plane is isometric to the 3-geometry. Let the vertices of the unit ball of the 3-geometry be v1 , . . . , v6 , in this order, with subscripts modulo 6. By Lemma 3.16 any absorbing angle ^xoy in the 3geometry must contain at least one of ^vi ovi+1 . It follows that the configuration
53
5 Solving the local Steiner problem in two dimensions
Figure 5.8: Shortening the configurations in Figure 5.2
{ox1 , . . . , ox5 } must be as in Figure 5.2. This configuration cannot be Steiner, since the tree can then be shortened (Figure 5.8). It follows that degree 5 Steiner points do not exist in any normed plane. Similarly, vertices in an SMT of degree at least 6 exist only if the plane is the 3geometry. Again considering absorbing angles, we find that vertices of degree at least 7 cannot exist, and of degree 6 must be as in Figure 5.2. Again, this vertex cannot be a Steiner point, since then the tree can be shortened (Figure 5.8). It remains to prove that the configurations in Figure 5.2 are terminal, i.e. we have to show that • {ov1 , . . . , ov5 } is an SMT of {o, v1 , . . . , v5 }, and • {ov1 , . . . , ov6 } is an SMT of {o, v1 , . . . , v6 } in the 3-geometry. We here present proofs which avoid the checking of many cases. We only consider the degree 6 configuration; there is a similar argument for the degree 5 configuration. For the sake of this argument we assume that all edges of SMTs are in one of the directions vi and we allow Steiner points of degree 2. Let r be the infimum of the radius ρ of a ball Bρ centered at o such that there exists an SMT of {o, v1 , . . . , v6 } such that the only edges outside Bρ are from vi ’s in the direction −vi . A simple limiting argument shows that the infimum is attained. We now show that r = 0. It is sufficient to show that any SMT of {o, v1 , . . . , v6 } can be modified such that for any i, there is only one edge connected to vi , pointing in the direction −vi . It is easily seen that only SMTs contained in the unit ball need be considered. Let v1 be connected to x 6∈ ov1 . If x ∈ int ^ov1 v2 , we can replace edge v1 x by two edges v1 p with direction −v1 and px with direction v3 , and still have an SMT. If v1 v2 is an edge, then we may replace it by either ov1 or ov2 , depending on whether the path from v1 to o passes through v2 or not. If x is a Steiner point of degree at least 3 on v1 v2 , then we may reroute around x, as in Figure 5.9. Since x has degree at least 3, the tree remains an SMT. Finally, if x is a Steiner point of degree 2 on v1 v2 , say x is connected to v1 and y, then we may “flip” the edges v1 x and xy, i.e. replace them by v1 z and zy, where z = v1 + y − x. The other cases are treated similarly. In the next section we extend the idea at the end of the above proof to give an alternative proof that star configurations with all angles absorbing are terminal in
54
5.3 Another approach for terminal configurations v2
x
v1
o Figure 5.9: Rerouting p2
p1
q1
q2
pk m qk
` s Figure 5.10
any Minkowski plane.
5.3 Another approach for terminal configurations In this section we give an alternative proof of (1) of Theorem 5.1 that a star {opi : i ∈ [k ]} in a normed plane is a Steiner minimal tree of its vertices {o, p1 , . . . , pk } if and only if all angles formed by the edges at o are absorbing. In the next section we also find a new sufficient condition for higher-dimensional normed spaces to share this characterization. This work was published in [73]. We first need the following technical result. Lemma 5.12. Let ` be a line passing through a Steiner point s of a SMT T in a Minkowski plane M2 . Assume that the norm is differentiable at non-zero vectors parallel to `. Then T has edges incident to s in both open half planes bounded by `. Proof. Without any assumption on `, the edges incident to s cannot all lie in the same open half plane bounded by `. Indeed, such a tree can be shortened as follows (Fig. 5.10). Let some line m intersect the interior of each edge spi , i ∈ [k ], in qi , say. Remove edges sp1 , spk , and sqi , i = 2, . . . , k − 1, and add edges p1 q2 , qi qi+1 , 2 ≤ i ≤ k − 2, and qk−1 pk , to obtain a new Steiner tree T 0 without the Steiner point s, but with new Steiner points qi , 2 ≤ i ≤ k − 1. By the triangle inequality, `( T ) − `( T 0 ) ≥ ∑ik=−21 ks − qi k > 0, contradicting the minimality of T. We now use the assumption on `. It is sufficient to show that sp1 and spk cannot be opposite edges both on `, with all other spi , 2 ≤ i ≤ k − 1, on the same side
55
5 Solving the local Steiner problem in two dimensions p2
p3
s2 s3 p1
p k −1
s k −1
pk
s Figure 5.11
of ` (Fig. 5.11). Let s2 be a variable point on sp2 with ks2 − sk small. Denote the intersection of s2 pk and spi by si , i = 3, . . . , k − 1. Change the Steiner tree T as follows. Remove edges p1 s, pk s and si s, i = 2, . . . , k − 1, and add edges p1 s2 and s2 pk . This removes the Steiner point s and introduces new Steiner points s2 , . . . , sk−1 . Denoting the new tree by T 0 , it follows that the length changes by
`( T 0 ) − `( T ) k −1
=ks2 − p1 k + ks2 − pk k − ks − p1 k − ks − pk k −
∑ k s − si k
i =2
≤k(s − p1 ) + (s2 − s)k − ks − p1 k + k(s − pk ) + (s2 − s)k − ks − pk k − ks2 − sk. Since s − p1 and s − pk are parallel to `, the norm is differentiable at these two points. Thus if ϕ is the norming functional of s − p1 then − ϕ is the norming functional of s − pk , and
k(s − p1 ) + (s2 − s)k − ks − p1 k s2 → s k s2 − s k k(s − pk ) + (s2 − s)k − ks − pk k = − lim . s2 → s k s2 − s k
ϕ(s2 − s) = lim
It follows that
`( T 0 ) − `( T ) ≤ ϕ(s2 − s) − ϕ(s2 − s) + o (ks2 − sk) − ks2 − sk = o (ks2 − sk) − ks2 − sk, which is negative if ks2 − sk is sufficiently small. Then `( T 0 ) < `( T ), a contradiction. Alternative proof of Theorem 5.1(1). Without loss of generality, the edges op1 , . . . , opk are ordered around o. Assume that all angles ^ pi op j are absorbing. We start off with an arbitrary SMT of {o, p1 , . . . , pk } and modify it in two steps without increasing the length. In Step 1 we eliminate all Steiner points in the interiors of the angles −→ ^ pi opi+1 . In Step 2 we eliminate all edges between vertices on different rays opi . The edges of the final SMT are then all contained in the the union of the segments opi , i = 1, . . . , k. This tree cannot have Steiner points, and so has to be the star with center o. This concludes the proof.
56
5.3 Another approach for terminal configurations pi
p i +1 qei
pei
ri o
Figure 5.12
Step 1: For each angle ^ pi opi+1 (where we let k + 1 ≡ 0), choose a regular −→ −−−→ direction ri not contained in the (closed) angle. Choose pei ∈ opi and qei ∈ opi+1 such that pei qei is parallel to ri (Fig. 5.12). For each point s in the interior of ^ pi op j , write s = αp + βq (uniquely, and then, moreover, α, β > 0) and define the measure of s to be |s| := α + β. Define the measure | T | of any Steiner tree T of {o, p1 , . . . , pk } to be the sum of the −→ measures of all Steiner points of T not on any ray opi . Let µ = inf{| T | : T is a SMT of {o, p1 , . . . , pk }}. Let Tn be a sequence of SMTs of {o, p1 , . . . , pk } with limn→∞ | Tn | = µ. Since there are only finitely many combinatorial types of Steiner trees on a set of k + 1 points, we may, by passing to a subsequence, assume without loss of generality that all Tn (n) (n) have the same combinatorial type with Steiner points s1 , . . . , sm , say. By taking further subsequences, we may assume that each sequence of Steiner points con(n) verge, say si → si , i ∈ [m]. In the limit we obtain a Steiner tree T0 with `( T0 ) = limn→∞ `( Tn ), hence T0 is a SMT. Also, | T0 | ≤ limn→∞ | Tn |, since the measure of a (n) Steiner point is continuous in the interior of an angle, hence limn→∞ |si | = |si | if (n)
si is still in the interior of the same angle, otherwise limn→∞ |si | ≥ |si | = 0 if si is −→ on one of the rays op j . Therefore, | T0 | = µ. It remains to show that µ = 0, since this will imply that T0 does not have any Steiner point in the interior of an angle. Suppose that µ > 0. We obtain a contradiction by constructing a SMT T 0 with | T 0 | < µ. Let s be a Steiner point of T0 in the interior of ^ pi opi+1 , say (Fig. 5.13(a)). Without loss of generality, no point of T0 is in the translated angle s + ^ pi opi+1 , since such a point is necessarily another Steiner point s0 and we may then repeatedly choose a new Steiner point s00 in s0 + ^ pi opi+1 , until this procedure halts. Let ` be the line through s parallel to ri . The points on ` in the interior of ^ pi opi+1 all have the same measure, and the points on the same side of ` as o have smaller measure. By Lemma 5.12 there is an edge sx1 incident to s on the same side of ` as o. There are at least two more edges sx2 and sx3 . Since not all edges are in an open half plane bounded by a line through s, we may choose x2 and x3 such that the angle ^x2 sx3 contains the translated angle s + ^ p1 op2 in its interior (exclduing s). It follows that there is a point s0 on sx1 sufficiently close to s such that ^x2 s0 x3 contains the translate s0 + ^ p1 op2 , and so is still absorbing (Figure 5.13(b)). We may therefore replace the edges sx2 , sx3 and ss0 by s0 x2 and s0 x3 without lengthening T0 ,
57
5 Solving the local Steiner problem in two dimensions pi
p i +1
pi
x3 s
` x1
p i +1 x3
s
x2 x1
ri
s0
o
o
(a)
(b)
x2
Figure 5.13
to obtain a new SMT T 0 . However, |s0 | < |s|, hence | T 0 | < | T | = µ, which gives the required contradiction. Step 2: Note that for any absorbing angle ^ pi op j ,
k pi − ok + k p j − ok + ko − ok ≤ k pi − p j k + k p j − p j k + ko − p j k, i.e., k pi − p j k ≥ k pi k. Suppose that the SMT T has an edge between two points on different segments, say between qi on opi and q j on op j . Without loss of generality, the unique path in T from o to qi passes through q j (otherwise qi and q j may be interchanged). Since ^qi oq j is absorbing, kqi − q j k ≥ kqi k. We can then replace the edge qi q j by oq j , without losing connectivity and without lengthening T. This process may be repeated until all edges are on the segments opi , which finishes Step 2.
5.4 Antipodality and higher dimensions Part (1) of Theorem 5.1 does not hold anymore in Minkowski spaces of dimension at least 3. Consider for example Cieslik’s space MdZ . In Chapter 4 we already showed that the star joining o to all 2d+1 − 2 vertices of the unit ball is not a SMT of these vertices when d ≥ 3. Neverthless, all the angles are absorbing. On the other hand, the characterization Theorem 5.1(1) extends to CL spaces (cf. Section 2.6), which include the Hanner spaces, in particular `1d and `d∞ . We first introduce some more notions. Two points on the unit sphere S(M) are called antipodal if there exist distinct parallel hyperplanes supporting the two points. Equivalently, unit vectors a and b are antipodal if and only if k a − bk = 2. Antipodality is a sufficient conditions for an angle to be absorbing, in the following sense. Lemma 5.13. If a and b are antipodal unit vectors in a Minkowski space, then ^aob is an absorbing angle.
58
5.4 Antipodality and higher dimensions Proof. Since the union of the segments oa and ob form a shortest path from a to b, these two segments form a SMT of {o, a, b}, hence ^aob is absorbing. The converse of the above lemma is not necessarily true, as the Euclidean norm shows. We call the unit ball of a Minkowski space Steiner antipodal if two points a and b on the boundary of the unit ball are antipodal whenever ^aob is absorbing. Theorem 5.14. Consider the following properties of a set { p1 , . . . , pk } of unit vectors in a Minkowski space Md . All angles ^ pi op j are absorbing.
(5.3)
All distances k pi − p j k = 2.
(5.4)
The star {opi : i ∈ [k ]} is a SMT of { p1 , . . . , pk }.
(5.5)
The star {opi : i ∈ [k ]} is a SMT of {o, p1 , . . . , pk }.
(5.6)
Then the implications (5.4)⇒(5.5)⇒(5.6)⇒(5.3) hold. Furthermore, (5.3) to (5.6) are equivalent if, and only if, the norm is Steiner antipodal. Proof. The implications (5.5)⇒(5.6)⇒(5.3) follow from Proposition 4.1. The implication (5.3)⇒(5.4) is obviously equivalent to the definition of Steiner antipodality. This leaves (5.4)⇒(5.5). Note that the given star is a Steiner tree of length k. It is sufficient to show that all Steiner trees have length ≥ k. However, note that the open unit balls centered at the pi are pairwise disjoint, since k pi − p j k = 2 for distinct i 6= j. Any Steiner tree will have to join each pi to the boundary of the unit ball with center pi . The part of the Steiner tree inside this ball must therefore have length at least 1. It follows that the length of any Steiner tree must be at least k. In order to apply this result, we need a characterization of Steiner antipodal norms in terms of duality. Proposition 5.15. The following are equivalent in any Minkowski space Md : The norm is Steiner antipodal.
(5.7)
The unit ball is a polytope and any two disjoint faces of
(5.8)
the dual unit ball are at distance > 1. Proof. (5.7)⇐(5.8) follows from Lemma 6.2 and the definition of Steiner antipodality. (5.7)⇒(5.8) follows upon noting that if a convex body is not a polytope, then there are disjoint exposed faces that are arbitrarily close to each other. What is important for our purposes is that CL spaces turn out to be Steiner antipodal. Proposition 5.16. All CL spaces are Steiner antipodal. Proof. Recall that the dual of a CL space is a CL space (Proposition 2.6). It is by Proposition 5.15 sufficient to show that any two disjoint faces F and G of the unit ball B of a CL space are at distance > 1. Suppose that F is contained in the facet F 0 . Then all vertices of B disjoint from F must lie in the opposite facet − F 0 . It follows that G ⊆ − F 0 , and F and G are therefore at distance 2.
59
5 Solving the local Steiner problem in two dimensions
60
Chapter
6
Applications to specific Minkowski planes In this chapter we demonstrate the utility of Theorem 5.1 from the previous chapter by presenting a list of examples that rederives various known results in a simple and uniform way.
6.1 The Euclidean plane The classical results [31] are regained by noting that critical angles are exactly the 120◦ angles, and absorbing angles are ≥ 120◦ angles.
6.2 Strictly convex planes In [1] it is shown that a degree four Steiner configuration in a strictly convex plane must form a cross. This is also true for terminal configurations. Proposition 6.1. A degree four terminal configuration in a strictly convex normed plane consists of two pairs of collinear segments. Proof. Immediate from the following lemma. Lemma 6.2. In a Minkowski plane with unit ball B, let A = ^xoy be an absorbing angle, and ` the line through yb parallel to x. Then ` ∩ int B ⊆ A. Proof. For any z ∈ ox we have ky − zk + kzk + k x − zk ≥ kyk + k xk, hence ky − z k ≥ k o − y k. The converse of the above lemma is false, even in the Euclidean plane. This can be seen by letting ^xoy be a right angle.
61
6 Applications to specific Minkowski planes b
ϕ2 ϕ1 + ϕ2 a
o
o ϕ1
Figure 6.1: The unit ball and dual unit ball of the elliptic X-plane with norm k·k1 + k·k2
6.3 Differentiable planes Differentiable planes cannot be X-planes, hence Steiner points have degree three, and terminals degree at most three, in accordance with [1]. If there existed a smooth X-plane M, then by Lemma 3.10, there would exist unit functionals ϕ1 , ϕ2 in the strictly convex dual M∗ , satisfying k ϕ1 + ϕ2 k ≤ 1, k ϕ1 − ϕ2 k ≤ 1. The triangle inequality would then give that k( ϕ1 + ϕ2 ) + ( ϕ1 − ϕ2 )k = 2 and k ϕ1 + ϕ2 k = k ϕ1 − ϕ2 k = 1. Strict convexity would then imply that ϕ1 + ϕ2 and ϕ1 − ϕ2 are linearly dependent, i.e., ϕ1 + ϕ2 = ±( ϕ1 − ϕ2 ), a contradiction.
6.4 Piecewise differentiable, elliptic X-planes In [1] it is shown that a large class of symmetric roundings of the rectilinear norm give X-planes. Their examples are analytically complicated. Conger [30] introduced √ a very simple example, namely the norm k·k1 + λk·k2 where 0 < λ ≤ 2 + 2. This norm is clearly elliptic. We now√use Theorem 5.1 to show that this norm defines an X-plane if and only if λ ≤ 2 + 2. By Lemma 2.1, the dual unit ball is the Minkowski sum B12 + λB22 , where B1 is the unit ball of the rectilinear plane, and B2 the unit ball of the Euclidean plane. The unit ball of the original norm does not have a particularly simple representation in general. However, it can be shown that the arc of √ the unit circle in each quadrant is a conic arc which is hyperbolic √ when 0 < λ < 2, parabolic when λ = 2, and elliptic when λ > 2. For example, when λ = 1, the unit ball of this norm has the simple form (| x | − 1)(|y| − 1) ≥ 1/2, | x |, |y| ≤ 1. Figure 6.2 depicts the unit ball and the dual unit ball. The functionals ϕ1 and ϕ2 are norming functionals of a and b, respectively. Since k ϕ1 + ϕ2 k∗ ≤ 1, it follows by Lemma 3.10 that ^aob is absorbing. By symmetry its supplements are also absorbing, which shows that this plane is an X-plane. By Theorem 5.1 the cross configuration is an SMT with a Steiner point of degree 4. It can be seen that these four angles are essentially the only absorbing angles whose supplements are also absorbing. √ The extreme case λ = 2 + 2 is shown in Figure 6.2. Here k ϕ1 + ϕ2 k∗ = 1, so
62
6.5 An almost smooth X-plane b
ϕ2
ϕ1 + ϕ2
a o
o
ϕ1
Figure √ 6.2: The unit ball and dual unit ball of the elliptic X-plane with norm k·k1 + (2 + 2)k·k2 ϕ2
b
ϕ1 + ϕ2 a o
o ϕ1
Figure 6.3: Almost smooth X-plane with dual this plane barely makes it to be an X-plane. For larger λ, the plane will not be an X-plane anymore. The Banach-Mazur distance between (R2 , k·k1 + λk·k2 ) equals √ √ √ 2+ λ 2 equals (2 + 4 2)/7 = 1.0938 . . . . This is not 1+λ , which in the case λ = 2 + the closest X-plane to the Euclidean plane. We will later see that the 6-geometry (with unit ball a regular dodecagon) is an X-plane. Its Banach-Mazur distance to the Euclidean plane is 1/ cos 15◦ = 1.035 . . . .
6.5 An almost smooth X-plane The boundary of the unit ball in the previous example has two antipodal pairs of non-smooth points. We now smooth this example still further, by slightly rounding two pairs of vertices of the unit ball of the 3-geometry (Figure 6.3). The dual unit ball is the Minkowski sum of an ellipse and a line segment parallel to the minor axis of the ellipse. Thus the dual unit ball is the Minkowski sum of an convex body and the unit ball of some Euclidean norm. By Lemma 2.1 it follows that the original norm is elliptic. Again, by considering the dual we see that this plane is an X-plane, hence degree 4 Steiner points exist. This example, with only one pair of antipodal non-smooth points, is essentially best possible since, as was seen in Example 6.3, differentiable planes cannot be X-planes.
63
6 Applications to specific Minkowski planes p2 ϕ2
p3 o
p1
ϕ3
ϕ10 ϕ1
o ϕ4
p4
Figure 6.4: An X-plane with a degree 4 terminal configuration with no collinear edges
Figure 6.5: Pointed degree 3 configurations with three critical angles that are not floating FT
6.6 A strange example The following example shows that no two edges of a degree four terminal configuration need to be collinear (as would happen with degree 4 Steiner configurations). Figure 6.4 shows an X-plane and its dual, with a degree four configuration with no two edges collinear. The unit ball is
B = {±
1 0
,±
0 1
,±
− 21 1
,±
−1 1 2
}.
The angle ^ p1 op2 is absorbing as demonstrated by the norming functionals ϕ1 , ϕ2 of p1 , p2 , respectively. Similarly, ϕ2 , ϕ3 show that ^ p2 op3 is absorbing, and ϕ3 , ϕ4 that ^ p3 op4 is absorbing. Finally, ^ p4 op1 is absorbing because of ϕ4 and ϕ10 . It follows by Theorem 5.1 that the configuration is terminal. By deleting either op3 or op4 from the configuration, we obtain a pointed configuration that is not a floating FT configuration, but with all three angles critical, showing that we need non-pointedness in the characterization of degree three Steiner points in the Theorem 5.1 (Figure 6.5).
64
6.7 The λ-geometry λ ≡ 0 (mod 3)
λ ≡ 1 (mod 3)
λ ≡ 2 (mod 3)
2λ −1 3
2λ−2 3
sides
sides
2λ −1 3
2λ−1 3
sides
2λ−1 3
sides
sides
2λ−2 3
sides
2λ −1 3
2λ−2 +1 3
2λ−1 3
sides
sides
sides
Figure 6.6: Degree 3 Steiner configurations in the λ-geometry
6.7 The λ-geometry 6.7.1 Degree three Steiner configurations In Section 3.3.2 we described the absorbing and critical angles of the λ-geometry, which also gave the floating FT configurations of degree 3. By Theorem 5.1, this also describes the Steiner configurations of degree 3. The following is then just a reformulation of Proposition 3.17. Proposition 6.3. A degree 3 Steiner configuration in the λ-geometry must be as in Figure 6.6. Note that if λ ≡ 2(mod 3) then all terminal configurations are already Steiner. In the other two cases this is not true, showing that we cannot replace critical angles by absorbing angles in the characterization of degree 3 Steiner configurations in the Theorem 5.1 (cf. also the remarks after Proposition 3.17).
6.7.2 Degree three terminal configurations By Lemma 3.16 each absorbing angle contains at least d 2λ 3 e − 1 full segments of the unit circle. By Theorem 5.1 we then obtain the following description of terminal configurations of degree 3. Proposition 6.4. A degree 3 terminal configuration in the λ-geometry must be as in Figure 6.7. Note that at least i edges of the configuration must pass through vertices of the unit circle, where λ ≡ i (mod 3), i = 0, 1, 2. Also note that the 3-geometry provides another example of a degree 3 pointed configuration that is not floating FT (cf. Figure 6.5 and Example 6.6).
6.7.3 Degree four Steiner and terminal configurations From Lemma 3.16 it follows that the λ-geometry is an X-plane if, and only if, 2(d 2λ 3 e − 1) ≤ λ. This inequality has solutions λ = 2, 3, 4, 6. By Theorem 5.1 it
65
6 Applications to specific Minkowski planes
λ ≡ 0 (mod 3) 2λ −1 3
2λ −1 3
2λ −1 3
sides
sides
sides
2λ −1 3
sides 2λ −1 3
sides
2λ −1 3
2λ −1 3
2λ −1 3
2λ −1 3
sides
sides
sides
sides
λ ≡ 1 (mod 3) 2λ−2 3
2λ−2 3
sides
sides
2λ−2 3
sides
2λ−2 3
2λ−2 3
2λ−2 3
sides
sides
sides
λ ≡ 2 (mod 3) 2λ−1 3
sides
2λ−1 3
A thickened side of the unit circle indicates
sides
that the edge passing through that side is free to move
2λ−1 3
sides
Figure 6.7: Degree 3 terminal configurations in the λ-geometry
66
6.7 The λ-geometry λ=2
λ=3
λ=4
λ=6
Figure 6.8: X-plane λ-geometries follows that it is only in these four λ-geometries that configurations of degree 4 can exist in SMTs, and they must be as in Figures 5.1 and 6.8. Here all degree 4 terminal configurations are also Steiner, except the one in Figure 5.1. Note that if λ = 3, there is a continuous family of degree 4 configurations at a fixed vertex. It is surprising that there is an X-plane (the 6-geometry) so close to the Euclidean plane. It can in fact be shown that the Banach-Mazur distance between any Xplane and the Euclidean plane cannot be smaller than the distance between the 6-geometry and the Euclidean plane.
6.7.4 Summary If we stipulate that all edges in an SMT in the λ-geometry must be in one of the λ directions determined by the vertices of the unit ball (cf. [91]), then we get the following characterization of terminal and Steiner configurations. Proposition 6.5. We consider SMTs in the λ-geometry with all edges in one of the λ directions determined by the vertices of the unit ball. 1. The three angles of a degree three terminal configuration must each contain exactly the following number of edges of a unit ball centered at the terminal. 2λ 2λ ( 2λ 3 , 3 , 3 ) or
λ ≡ 0(mod 3)
2λ 2λ ( 2λ 3 + 1, 3 , 3 − 1) or 2λ 2λ ( 2λ 3 + 2, 3 − 1, 3 − 1)
λ ≡ 1(mod 3)
( 2λ3−2 + 2, 2λ3−2 , 2λ3−2 ) or
λ ≡ 2(mod 3)
( 2λ3−1 + 1, 2λ3−1 , 2λ3−1 )
( 2λ3−2 + 1, 2λ3−2 + 1, 2λ3−2 )
67
6 Applications to specific Minkowski planes 2. The three angles of a degree three Steiner configuration must each contain exactly the following number of edges of a unit ball centered at the Steiner point. λ ≡ 0(mod 3)
2λ 2λ ( 2λ 3 , 3 , 3 ) or 2λ 2λ ( 2λ 3 + 1, 3 , 3 − 1)
λ ≡ 1(mod 3)
( 2λ3−2 + 1, 2λ3−2 + 1, 2λ3−2 )
λ ≡ 2(mod 3)
( 2λ3−1 + 1, 2λ3−1 , 2λ3−1 )
3. The four angles of a degree four terminal configuration must each contain exactly the following number of edges of a unit ball with center the terminal. λ=2 λ=3 λ=4 λ=6
(1, 1, 1, 1) (1, 1, 1, 3) or (1, 2, 1, 2) or (1, 2, 2, 1) (2, 2, 2, 2) (3, 3, 3, 3)
4. The four angles of a degree four Steiner configuration must each contain exactly the following number of edges of a unit ball with center the Steiner point. λ=2 λ=3 λ=4 λ=6
(1, 1, 1, 1) (1, 2, 1, 2) or (1, 2, 2, 1) (2, 2, 2, 2) (3, 3, 3, 3)
Note that our description of degree three Steiner configurations in the case λ ≡ 0(mod 3) is at variance with [91] — we have ( 2λ3−1 + 1, 2λ3−1 , 2λ3−1 ) as a further possibility for a degree three Steiner point.
68
Chapter
7
An abstract solution in higher dimensions In this chapter we develop a general method of proving that certain star configurations in finite-dimensional normed spaces are Steiner minimal trees. This method generalizes the results of Lawlor and Morgan [63] that could only be applied to differentiable norms. The generalization uses some results from the subdifferential calculus (Section 2.7). We next introduce and study a binary operation defined on subsets of a Minkowski space, closely related to Minkowski addition, that we call reduced Minkowski addition. This operation is used in the formulation of the characterization of Steiner and terminal configurations in a Minkowski space (Theorems 7.2 and 7.3). After the proofs we reformulate the characterizations for strictly convex spaces [63, 96].
7.1 Reduced Minkowski addition We define the reduced Minkowski sum of two closed, convex subsets C and D of the unit ball of (Md )∗ to be C D = { ϕ + ψ : ϕ ∈ C, ψ ∈ D, k ϕ + ψk∗ ≤ 1}, i.e. C D is the usual Minkowski sum C + D intersected by the unit ball B∗ of (Md )∗ . Reduced Minkowski addition is clearly commutative but not associative, although it satisfies a weak associative law (see Proposition 7.1 below). We now consider some elementary properties of this binary operation. Consider a finite non-empty family Σ = { Ai : i ∈ I } of operands, where I ⊆ [k ] and each Ai is a closed convex subset of B∗ . We call I the support of Σ. A parenthesization of Σ is defined to be a parenthesization, in the usual sense, of some ordering A j (1) . . . A j ( i )
(7.1)
of Σ, where j : [| I |] → I is some bijection. Stanley calls this a binary set bracketing ¨ of Σ [95, p. 178], and its enumeration is called Schroder’s third problem [93]. We
69
7 An abstract solution in higher dimensions denote a parenthesization of Σ by hΣi, and we also call the family I the support of hΣi. We may define a parenthesization of Σ recursively as follows. • If Σ is the singleton { Ai }, then hΣi = Ai is the only parenthesization of Σ. • If Σ = Σ1 ∪ Σ2 , with Σ1 and Σ2 disjoint, and hΣi i is a parenthesization of Σi for each i = 1, 2, then both hΣi = (hΣ1 i hΣ2 i) and hΣi = (hΣ2 i hΣ1 i) are equivalent parenthesizations of Σ. More generally, two parenthesizations of Σ are equivalent if they can be transformed into each other using the commutative law on any subexpression. This is an equivalence relation, and two equivalent parenthesizations clearly evaluate to the same set. The collection of equivalence classes of parenthesizations of a set Σ with support I corresponds bijectively with the collection of abstract trees with | I | + 1 leaves labelled by the elements of {0} ∪ I, and with | I | − 1 internal vertices of degree 3. We call such a tree a rooted abstract Steiner tree on I. The node 0 is the root of the tree, node i corresponds to Ai for each i ∈ I, and each internal vertex corresponds to an instance of in the parenthesization corresponding to the tree. Note that we do not distinguish between left and right branches (as is done in the well-known bijection between parenthesizations of ordered expressions and planted, trivalent plane trees, both counted by the Catalan numbers; see [95, Exercise 6.19(b),(f)]). Stanley calls this a binary total partition [95, Example 5.2.6] of the set I. The explicit construction of the bijection for our variant is the following. We denote the rooted abstract Steiner tree associated to hΣi by T0 hΣi. • If hΣi = Ai then T0 hΣi is the tree joining 0 and i. • If hΣi = (hΣ1 i hΣ2 i) then T0 hΣi is the tree obtained by identifying the vertices 0 in T0 hΣ1 i and T0 hΣ2 i to a single vertex x (corresponding to the instance of operating on hΣ1 i and hΣ2 i), and joining x to a new root 0. It is clear that two parenthesizations of Σ are equivalent if and only if their associated rooted abstract Steiner trees are equal. ¨ The solution to Schroder’s third problem, i.e., the number ak of equivalence classes of parenthesizations with support [k ], is the product of the first k − 1 odd numbers: k −1
ak =
∏ (2i − 1). i =1
This is seen as follows. Since ak equals the number of rooted abstract Steiner trees on [k ], i.e., trees with k + 1 leaves {0, 1, . . . , k } and k − 1 internal vertices of degree 3, we have a1 = 1 and ak+1 = (2k − 1) ak , since we may subdivide any of the 2k − 1 edges of such a tree on [k ] and join the new vertex to k + 1, to obtain such a tree on [ k + 1]. We remark that the kth Catalan number can be derived from ak . Since there are k − 1 ’s in a parenthesization of { Ai : i ∈ [k ]}, there are 2k−1 parenthesizations in an equivalence class. This gives 2k−1 ak parenthesizations (in the usual sense) of the ordered expression Aπ (1) . . . Aπ (k) , where π is a permutation of [k ], taken over all permutations π. Therefore, the number of parenthesizations of A1 . . . Ak
70
7.2 The characterization is 2k−1 ak /k!, which equals the kth Catalan number. This derivation of the Catalan numbers is essentially the same as the classical combinatorial derivation of Rodrigues [87]. We define an abstract Steiner tree on I to be a tree with set of leaves I, and with | I | − 2 internal vertices of degree 3. The abstract Steiner tree associated to a rooted abstract Steiner tree on I is obtained by contracting the root 0. The abstract Steiner tree of a parenthesization hΣi obtained in this way from T0 hΣi is denoted by T hΣi. The number of abstract Steiner trees on [k ] clearly equals ak−1 . We call two parenthesizations weakly equivalent if they can be transformed into each other using the commutative law on any subexpression or the associative law on the whole expression, i.e., a bracketed expression
((hΣ1 i hΣ2 i) hΣ3 i) may be transformed into
(hΣ1 i (hΣ2 Σ3 i)), and vice versa. It is again clear that two parenthesizations Σ and Σ0 are weakly equivalent if and only if their abstract Steiner trees are equal: T hΣi = T hΣ0 i. It follows that the number of weak equivalence classes of parenthesizations on [k ] equals ak−1 . The operation has the following weak associativity property. Proposition 7.1. Let hΣi1 and hΣi2 be two weakly equivalent parenthesizations of Σ = { Ai : i ∈ I }, where each Ai is a closed convex subset of B(M∗ ). Then o ∈ hΣi1 if and only if o ∈ hΣi2 . Proof. By the definition of weak equivalence, it is sufficient to show for any three Ai , i = 1, 2, 3, that o ∈ h A1 i (h A2 i h A3 i) if and only if o ∈ (h A1 i h A2 i) h A3 i. This is clear.
7.2 The characterization Theorem 7.2 (Nodes). Let N = { p0 , p1 , . . . , pk } be a set of points in a Minkowski space Md . Then the star joining p0 to each pi , i ∈ [k ], is an SMT of N if and only if hΣi 6= ∅ for each parenthesization hΣi of Σ = {∂( pi − p0 ) : i ∈ [k ]}. Proof. ⇒: Consider any parenthesization hΣi of Σ and its associated rooted abstract Steiner tree T0 hΣi. We now turn this tree into a Steiner tree in Md . Associate leaf i with pi for each i ∈ {0} ∪ [k ], and associate each of the k − 1 internal vertices with a variable point xi ∈ Md , i ∈ [k − 1]. Denote this Steiner tree by T0 hΣi( x1 , . . . , xk−1 ). Its length is L( x1 , . . . , xk−1 ) = `( T0 hΣi( x1 , . . . , xk−1 )). The Steiner points xi in T0 hΣi( x1 , . . . , xk−1 ) may coincide — this results in the tree being in fact a contraction of T0 hΣi. Note that L : Md × · · · × Md → R is a convex function since L ( x 1 , . . . , x k −1 ) = ∑ ρ e ( x 1 , . . . , x k −1 ), e∈ E( T0 hΣi)
71
7 An abstract solution in higher dimensions where ρe ( x1 , . . . , xk−1 ) = k a − bk, with a and b the two points in N ∪ { x1 , . . . , xk−1 } associated to the vertices incident to e. Each ρe depends on only one or two of the variables xi . Since T0 hΣi( p0 , . . . , p0 ) is the star joining p0 to all pi , i ∈ [k ], which is an SMT by assumption, L attains its minimum at ( p0 , . . . , p0 ). Thus o ∈ ∂L( p0 , . . . , p0 ) =
∑
∂ρe ( p0 , . . . , p0 )
(7.2)
e∈ E( T0 hΣi)
by Lemma 2.8. By Lemmas 2.9, 2.10 and 2.11, if e = xi x j , then ∂ρe ( p0 , . . . , p0 ) = {(o, . . . , o, ϕ , o, . . . , o, − ϕ , o, . . . , o ) : ϕ ∈ B∗ } ↑ ↑ position i
position j
while if e = pi x j , then ∂ρe ( p0 , . . . , p0 ) = {(o, . . . , o, ϕ , o, . . . , o ) : ϕ ∈ ∂( pi − p0 )}. ↑ position j
By considering each coordinate i ∈ [k − 1] of (7.2) we obtain a functional ϕe ∈ ( Md )∗ for each edge e ∈ E( T0 hΣi) such that ( B∗ if e = xi x j or e = p0 xi , • ϕe ∈ ∂( pi − p0 ) if e = pi x j , i = 1, . . . , k − 1, • and for each Steiner point xi , ϕe = ϕ f + ϕ g , where e is the incoming edge and f , g the two outgoing edges of xi , when the tree is directed away from the root p0 . By induction on the definition of T0 hΣi (≡ induction on subexpressions of hΣi) we obtain that ϕe ∈ hΣi, where e = p0 xi is the root edge. This gives hΣi 6= ∅. ⇐: Consider any Steiner tree in Md on { p0 , . . . , pk }. By subdividing points if necessary, we obtain a tree with leaves { p0 , . . . , pk } and with k − 1 Steiner points xi of degree 3, some of them possibly coinciding with each other or with the pi . This tree is the rooted abstract Steiner tree of some parenthesization hΣi of Σ. As in the “⇒”-argument, we obtain that hΣi 6= ∅ implies that o ∈ ∂L( p0 , . . . , p0 ), i.e., L attains its minimum at ( p0 , . . . , p0 ), which is when the tree is T0 hΣi( p0 , . . . , p0 ), the star joining p0 to the other pi . Theorem 7.3 (Steiner points). Let N = { p1 , . . . , pk } be a set of points in a Minkowski space Md . Then the star joining p0 ∈ / N to each pi , i ∈ [k ], is an SMT of N if and only if o ∈ hΣi for each parenthesization hΣi of Σ = {∂( pi − p0 ) : i ∈ [k ]}, if and only if o ∈ hΣ0 i + ∂( pk − p0 ) for each parenthesization hΣ0 i of Σ0 = {∂( pi − p0 ) : i ∈ [k − 1]}. Proof. Since any parenthesization hΣi is weakly equivalent to hΣ0 i ∂( pk − p0 ) for some parenthesization of hΣ0 i, Proposition 7.1 gives the equivalence between the two conditions. We next show that the first condition is necessary and sufficient.
72
7.2 The characterization
⇒: This is similar to the proof of Theorem 7.2. Consider any parenthesization hΣi of Σ. Turn its associated rooted abstract Steiner tree T0 hΣi into an abstract Steiner tree T hΣi by contracting the root. We turn T hΣi into a directed graph as follows. Denote the (new) edge into which the root was contracted by e, and give it both directions, denoting the two directed edges by e+ and e− . Give all other edges of T hΣi a single direction away from e. This tree becomes a Steiner tree in Md as follows. Associate leaf i with pi for each i ∈ [k ], and associate each of the k − 2 internal vertices with a variable point xi ∈ Md , i ∈ [k − 2]. Denote this Steiner tree by T hΣi( x1 , . . . , xk−2 ). Its length is the convex function L( x1 , . . . , xk−2 ) = `( T hΣi( x1 , . . . , xk−2 )). Again note that the Steiner points xi in T hΣi( x1 , . . . , xk−2 ) may coincide, and then the tree is a contraction of T hΣi. Again, L attains its minimum at ( p0 , . . . , p0 ). Calculating the subdifferential coordinate-wise, we obtain a functional ϕ~e ∈ ( Md )∗ for each directed edge ~e of E( T hΣi) such that ( B∗ if ~e is incident with two Steiner points, • ϕ~e ∈ ∂( pi − p0 ) if ~e is incident with pi , i = 1, . . . , k, • ϕe+ = − ϕe− , and • for each Steiner point xi , ϕ~e = ϕ~f + ϕ~g , where ~e is the incoming edge and ~f , ~g the two outgoing edges of xi , with the convention that we ignore the outgoing e+ or e− if xi is incident with e. Write hΣi = hΣ+ i hΣ− i, where e± points to the subtree associated with hΣ± i. Let I ± be the support of hΣ± i, and for each i ∈ [k ] let ϕi = ϕ~e , where ~e is incident with xi . Again by induction on subexpressions we obtain that ϕe± = ∑i∈ I ± ϕi ∈ hΣ± i. From ϕe+ = − ϕe− it follows that o ∈ hΣi. ⇐: Similar to the corresponding direction in the proof of Theorem 7.2. When applying Theorem 7.2 (Theorem 7.3) there are ak (ak−1 , respectively) parenthesizations to consider. Note that in the “⇐”-directions of the above proofs we did not need the parts of Lemmas 2.7 to 2.11 depending on the separation theorem, i.e., Lemma 2.7 and the “⊆”-part of Lemma 2.8. These directions of Theorems 7.2 and 7.3 are the ones used to obtain lower bounds for τ (Md ) and σ(Md ). On the other hand, to obtain upper bounds we need the “⇒”-directions, where the separation theorem is needed. Recall that if the norm is differentiable, then ∂k xk is a singleton whenever x 6= o. This drastically simplifies the conditions in Theorems 7.2 and 7.3 and we regain the following two results. Corollary 7.4 (Lawlor and Morgan [63]). Let N = { p1 , . . . , pn } be a set of points, all different from the origin o, in a Minkowski space M with differentiable norm. Let ϕi be the norming functional of pi , i = 1, . . . , n. Then the star joining o to each pi is a SMT of N if and only if n
∑ ϕi = o
i =1
73
7 An abstract solution in higher dimensions and for each subset I ⊆ {1, . . . , n},
k∑ ϕk∗ ≤ 1. i∈ I
Corollary 7.5 ( [96]). Let N = { p1 , . . . , pn } be a set of points, all different from the origin o, in a Minkowski space M with differentiable norm. Let ϕi be the norming functional of pi , i = 1, . . . , n. Then the star joining o to each pi is a SMT of {o} ∪ N if and only if for each subset I ⊆ {1, . . . , n}, k∑ ϕk∗ ≤ 1. i∈ I
74
Chapter
8
Applications to specific high-dimensional spaces In this chapter we apply Theorems 7.2 and 7.3 from the previous chapter to two special Minkowski spaces. The first is Cieslik’s MdZ . We determine the maximum degree of a terminal in a Steiner minimal tree τ (MdZ ). The proof makes essential use of extremal finite set theory. Then we give an estimate for the maximum degree of a Steiner point σ(MdZ ) by providing an asymptotically matching lower bound. The second space, occurring in the work of Mark Conger [30], has the elliptic norm that equals the sum of the `1 -norm and a small multiple of the `2 norm. For this norm we determine the maximum degree of a Steiner point.
8.1 The maximum degree of a terminal in MdZ d +2 Theorem 8.1. For any d ≥ 2, τ (MdZ ) = (b(d+ 2)/2c).
In order to apply Theorem 7.2, we need to describe the norming functionals. Lemma 8.2. For any non-zero x ∈ MdZ , we have ∂x = F 0 ( X ), where X ∈ Zd is the unique signed set such that F ( X ) is the face of Zd wich has xb in its relative interior. This follows from the discussion in Section 2.5. We thus have to determine for which families Σ = { Xi : i ∈ [k ]} ⊆ Zd all parenthesizations h F 0 (Σ)i of F 0 (Σ) := { F 0 ( Xi ) : i ∈ [k]} are non-empty. To this end, we define a commutative, nonassociative binary operation on Zd as follows: • X
1=1
X = X for all X ∈ Zd , and
• for all X, Y 6= 1,
X
(X+ , Y− ) (Y + , X − ) Y := 1 ∅
if X + ∩ Y − if X + ∩ Y − if X + ∩ Y − if X + ∩ Y −
= ∅ and X − ∩ Y + 6= ∅ and X − ∩ Y + 6= ∅ and X − ∩ Y + = ∅ and X − ∩ Y +
6= ∅, = ∅, 6= ∅, = ∅.
75
8 Applications to specific high-dimensional spaces Extending F 0 to Zd by defining F 0 (1) = {0}, we obtain Lemma 8.3. For all X, Y ∈ Zd , F0 (X
Y ) ⊆ F 0 ( X ) F 0 (Y ) .
Furthermore, if at least one of the conditions X = 1, Y = 1, X + ∩ Y − = ∅ or X − ∩ Y + = ∅ holds, then F 0 ( X Y ) = F 0 ( X ) F 0 (Y ) . Proof. We assume that X, Y 6= 1, otherwise the lemma is trivial. Consider the first relation. Since this is trivial if X Y = ∅, we assume that X and Y are not conformal, say that X + ∩ Y − 6= ∅. There are now two cases, depending on whether X − ∩ Y + is empty or not. • If X − ∩ Y + = ∅, we have to show that F 0 (Y + , X − ) ⊆ F 0 ( X ) F 0 (Y ). Choose any i ∈ X + ∩ Y − . Then for any ϕ ∈ F 0 (Y + , X − ) we have ϕ(i ) = 0. By setting ψ = ϕ− + 12 ei ∈ F 0 ( X ) and χ = ϕ+ − 12 ei ∈ F 0 (Y ), we obtain ϕ = ψ + χ ∈ F 0 ( X ) F 0 (Y ) . • If X − ∩ Y + 6= ∅, we have to show that F 0 (1) ⊆ F 0 ( X ) F 0 (Y ). Choose any i ∈ X + ∩ Y − and j ∈ X − ∩ Y + . Then 12 ei − 12 e j ∈ F 0 ( X ) and − 12 ei + 12 e j ∈ F 0 (Y ), giving o ∈ F 0 ( X ) F 0 (Y ). This establishes the inclusion. Now consider the second part of the lemma. Without loss of generality we may assume that X + ∩ Y − = ∅. Let ϕ ∈ F 0 ( X ) and ψ ∈ F 0 (Y ) be such that k ϕ + ψk1 ≤ 1. Then for each i ∈ X + we have 0 ≤ ψ(i ) ≤ ϕ(i ) + ψ(i ) and for each i ∈ Y − we have 0 ≥ ϕ(i ) ≥ ϕ(i ) + ψ(i ). In particular, X + ∩ supp− ( ϕ + ψ) = ∅ and Y − ∩ supp+ ( ϕ + ψ) = ∅, and 1 ≥ k ϕ + ψ k1
∑
=
( ϕ(i ) + ψ(i )) +
≥
∑
−( ϕ(i ) + ψ(i ))
i ∈supp− ( ϕ+ψ)
i ∈supp+ ( ϕ+ψ)
∑ ( ϕ(i) + ψ(i)) + ∑− −( ϕ(i) + ψ(i))
i∈X +
i ∈Y
1 1 = + ∑ ψ (i ) + + ∑ − ϕ (i ) 2 i∈X + 2 i ∈Y −
≥ 1. Thus equality holds everywhere, giving k ϕ + ψk1 = 1, supp+ ( ϕ + ψ) ⊆ X + , and supp− ( ϕ + ψ) ⊆ Y − . Therefore, ϕ + ψ ∈ F 0 ( X + , Y − ). If X − ∩ Y + 6= ∅, then ( X + , Y − ) = X Y, and we are done. Otherwise, X − ∩ Y + = ∅, giving that X and Y are conformal. Then F 0 ( X Y ) = ∅, and also F 0 ( X ) F 0 (Y ) = ∅, since F 0 ( X ) and F 0 (Y ) belong to the same facet of Zd∗ . Theorem 8.4. Let Σ = { Xi : i ∈ [k ]} ⊆ Zd \ {∅}. Then the following are equivalent. a. Some parenthesization of F 0 (Σ) is empty.
76
8.1 The maximum degree of a terminal in MdZ b. Some parenthesization of Σ (with operation
) equals ∅.
c. There exist indices a, b, c, d ∈ [k ] with { a, b} ∩ {c, d} = ∅ such that ( Xa+ ∪ Xc+ ) ∩ ( Xb− ∪ Xd− ) = ∅. Proof. The implication (a) =⇒ (b) follows from Lemma 8.3, since F 0 ( X ) = ∅ implies X = ∅. (b) =⇒ (c): Let hΣi = ∅. By the definition of , for some subexpression hΣ1 i hΣ2 i of hΣi we have hΣ1 i 6= ∅ 6= hΣ2 i but hΣ1 i hΣ2 i = ∅. If both Σ1 and Σ2 are singletons, say Σ1 = { Xi } and Σ2 = { X j }, then Xi and X j are conformal, i.e., Xi+ ∩ X j− = ∅ = X j+ ∩ Xi− , so we may take a = b = i and c = d = j. If Σ1 is a singleton { Xa } but Σ2 is not, then by the definition of there exist Xc , Xd ∈ Σ2 such that hΣ2 i = ( Xc+ , Xd− ) is conformal with Xa , i.e., Xa+ ∩ Xd− = ∅ = Xc+ ∩ Xa− . Thus we may set b = a. A similar argument takes care of the case where Σ2 is a singleton. If Σ1 and Σ2 are both not singletons, then by the definition of there exist − + Xa , Xb ∈ Σ1 and Xc , Xd ∈ Σ2 such that hΣ1 i = ( Xa , Xb ) and hΣ2 i = ( Xc+ , Xd− ) are conformal, giving (c). (c) =⇒ (a): It is sufficient to find a parenthesization of a subfamily of F 0 (Σ) that equals the empty set. Without loss assume that F 0 ( Xa ) F 0 ( Xb ) 6= ∅ 6= F 0 ( Xc ) F 0 ( Xd ). By Lemma 8.3 we have F 0 ( Xa ) F 0 ( Xb ) = F 0 ( Xa Xb ), hence Xa Xb 6= ∅, and Xa Xb = ( Xa+ , Xb− ). Similarly, F 0 ( Xc ) F 0 ( Xd ) = F ( Xc+ , Xd− ), and again by Lemma 8.3,
( F 0 ( Xa ) F 0 ( Xb )) ( F 0 ( Xc ) F 0 ( Xd )) = F 0 ( Xa+ , Xb− ) F 0 ( Xc+ , Xd− ) = F 0 (∅) = ∅. The above theorem together with Theorem 7.2 now gives the following. Corollary 8.5. Let P = { pi : i ∈ [k ]} be a family of points in MdZ \ {o}, with pi in the relative interior of face F ( Xi ) of Zd . Then the star connecting P to o is an SMT of P ∪ {o} if and only if there do not exist indices a, b, c, d ∈ [k ] with { a, b} ∩ {c, d} = ∅ and ( Xa+ ∪ Xc+ ) ∩ ( Xb− ∪ Xd− ) = ∅. Consequently, the star connecting P to o is an SMT of P ∪ {o} if, and only if, for each choice of indices a, b, c, d ∈ [k ], the star connecting { p a , pb , pc , pd } to o is a SMT of { p a , pb , pc , pd , o}. Corollary 8.6. In the Minkowski space MdZ , the star joining o to any set A of non-zero points is an SMT of A ∪ {o} if and only the star joining o to any subset B of A of size at most 4 is an SMT of B ∪ {o}. The problem of determining d(MdZ ) has now been reduced to a problem in extremal finite set theory. Theorem 8.7. Let { Xi : i ∈ [k ]} be a family of signed sets from Zd such that there do not exist indices a, b, c, d ∈ [k ] with { a, b} ∩ {c, d} = ∅ and ( Xa+ ∪ Xc+ ) ∩ ( Xb− ∪ Xd− ) = ∅. d +2 + 0 Then k ≤ (b(d+ 2)/2c) with equality if and only if all Xi = ∅, and either all | Xi | ∈ {b(d + 1)/2c, b(d + 1)/2c + 1}, or all | Xi+ | ∈ {d(d + 1)/2e − 1, d(d + 1)/2e}.
77
8 Applications to specific high-dimensional spaces Proof. It is easily seen that the hypothesis is equivalent to the following statement: For all families of sets {Yi : i ∈ [k ]} ⊆ P [d + 1] \ {∅, [d + 1]} such that Xi+ ⊆ Yi ⊆ [d + 1] \ Xi− , there do not exist indices a, b, c, d ∈ [k ] (8.1) with { a, b} ∩ {c, d} = ∅ and Ya ∪ Yc ⊆ Yb ∩ Yd . Property (8.1) is in turn equivalent to the following three conditions: all Yi are distinct,
(8.2)
there do not exist distinct a, b, c ∈ [k ] with Ya ⊆ Yb ⊆ Yc , and
(8.3)
there do not exist distinct a, b, c, d ∈ [k ] with Ya ∪ Yb ⊆ Yc ∩ Yd .
(8.4)
By a well-known generalization of Sperner’s theorem due to Erd˝os [2, 39], condid +2 tions (8.2) and (8.3) on their own already give the sharp upper bound k ≤ (b(d+ 2)/2c) with equality exactly when {Yi : i ∈ [k ]} consists of two of the largest levels in P [d + 1]. Because of this rigidity, it easily follows that in the case of equality all Xi0 = ∅, finishing the proof. This finishes the proof of Theorem 8.1. We remark that conditions (8.2) and (8.4) on their own give the same upper bound, but with an extra case of equality when d + 1 = 4; see [32].
8.2 A geometric formulation of Sperner’s theorem We now give a lower bound for σ(MdZ ) that asymptotically matches the upper bound of Theorem 8.1. √ d +1 d d Theorem 8.8. For any d ≥ 3, σ(MdZ ) ≥ (b(d+ 1)/2c). Therefore, σ (MZ ) = Θ (2 / d ). Proof. We only have to prove the lower bound. For this we need the next lemma, which follows from an observation of Moore [44]. Lemma 8.9. If the unit ball of Md contains k points on its boundary such that the distance between any two equals 2, then s(Md ) ≥ k. From this lemma together with Lemma 4.4 it immediately follows that s(`d∞ ) = d(`d∞ ) = 2d and s(`1d ) = d(`1d ) = 2d. We apply this lemma to MdZ . Recall that a vertex of Zd equals some F ( X ), where X 0 = ∅ and X ± 6= ∅. Thus X can be identified with the set X + ⊆ [d + 1], not equal to ∅ or [d + 1]. Then for any distinct X + , Y + ∈ P [d + 1] \ {∅, [d + 1]}, ( 1 if X + ⊂ Y + or Y + ⊂ X + , + + k F ( X ) − F (Y )k Z = 2 if X + 6⊆ Y + and Y + 6⊆ X + . It follows that a set { F ( Xi ) : i ∈ I } of vertices of Zd are all at pairwise distance 2 if and only if { Xi+ : i ∈ I } is an antichain. It follows from Lemma 8.9 that d +1 s(MdZ ) ≥ (b(d+ 1)/2c). This establishes the theorem.
78
8.3 Perturbations of `1d By Sperner’s theorem [2] we cannot do better: The largest such set of vertices d +1 has size (b(d+ 1)/2c). The same bounds are obtained when we consider arbitrary points on the boundary of Zd . For any boundary point p of Zd its support X p := supp( p) describes the unique face F ( X p ) which contains p in its relative interior. Two boundary points p and q are at distance 2 if and only if there exist parallel supporting hyperplanes at p and q with Zd inbetween. This in turn is equivalent to p and −q being contained in the same facet F ( X ) of Zd . Note that F ( X ) is a facet − of Zd if and only if | X ± | = 1. Therefore, k p − qk Z = 2 if and only if X + p ∩ Xq 6 = ∅ + − and X p ∩ Xq 6= ∅. Now let P = { pi : i ∈ I } be a set of boundary points of Zd at pairwise distance 2, let Xi = supp( pi ), and choose Yi ∈ P [d + 1] \ {∅, [d + 1]} such that Xi+ ⊆ Yi ⊆ [d + 1] \ Xi− . Then it follows that the Yi are all distinct sets, and d +1 form an antichain. As before we have by Sperner’s theorem that | P| ≤ (b(d+ 1)/2c), + with equality if and only if all pi are vertices and all Xi have the same cardinality, either b(d + 1)/2c or d(d + 1)/2e. We have shown the following. Proposition 8.10. The largest number of unit vectors in MdZ at pairwise distance 2 is d +1 (b(d+ 1)/2c). Even more general would be the following. d +1 Conjecture 8.11. The largest size of an equilateral set in MdZ equals (b(d+ 1)/2c).
This conjecture is known to hold for d = 3 [94] (and is easy for d = 2).
8.3 Perturbations of `1d Theorem 8.12. Let λ√> 0,√ and let Md = (Rd , k·k1 + λk·k2 ). If λ ≤ 1 then σ(Md ) ≤ d τ (M ) ≤ 2d. If λ ≤ d/( d − 1) then τ (Md ) ≥ σ(Md ) ≥ 2d. Proof. Let k·k = k·k1 + λk·k2 . By Lemma 2.1 we may identify the dual unit ball B∗ with ∑id=1 [−ei , ei ] + λB2∗ . Recall that each ∂k xk, x 6= o, is an exposed face of B∗ . Any exposed face of B∗ equals a proper non-empty face F of the cube ∑id=1 [−ei , ei ] translated by a functional ϕ with k ϕk2∗ = λ, such that F and ϕ have the same sign in the following sense: if F = ∑i∈X + ei − ∑i∈X − ei + ∑i∈X0 [−ei , ei ] for some signed set X 6= ∅, then X = supp( ϕ). We first show τ (Rd , k·k) ≤ 2d if λ ≤ 1. Let p1 , . . . , pk ∈ Md , pi 6= o, and suppose that the star joining o to all pi is an SMT of {o} ∪ { pi : i ∈ [k ]}. Then by Theorem 7.2 all restricted Minkowski sums of the ∂k pi k must be non-empty. Let ∂k pi k = Fi + ϕi as above, with Xi the corresponding signed set. Suppose k > 2d. Since all Xi 6= ∅, it follows from the pigeon-hole principle that for some two indices i, j, Xi and X j have a common element of the same sign, say 1 ∈ X1+ ∩ X2+ . Then, since F1 and F2 are both contained in the hyperplane {χ : χ(e1 ) = 1}, it follows that ∂k p1 k + ∂k p2 k is contained in the hyperplane {χ : χ(e1 ) = 2 + ϕ1 (e1 ) + ϕ2 (e1 )}. However, B∗ is contained in the slab bounded by ±{χ : χ(e1 ) = 1 + λ}. Since 2 + ϕ1 (e1 ) + ϕ2 (e1 ) > 2 ≥ 1 + λ, we obtain ∂k p1 k ∂k p2 k = ∅, a contradiction. Therefore, k ≤ 2d.
79
8 Applications to specific high-dimensional spaces
√ √ We now prove that σ(Md ) ≥ 2d for λ ≤ d/( d − 1) by showing that the star joining o to all ±ei , i ∈ [d], is an SMT of {±ei : i ∈ [d]}. This is trivial for d = 1, so we assume from now on that d ≥ 2. We have d
∂kei k = Ei :=
∑ [−ej , ej ] + (1 + λ)ei .
j =1 j 6 =i
By Theorem 7.3 it is sufficient to prove that for each parenthesization hΣi of Σ := {± Ei : i ∈ [d − 1]} ∪ { Ed } we have o ∈ hΣi − Ed , or equivalently, Ed ∩ hΣi 6= ∅. Write hΣi = hΣ1 i hΣ2 i = (hΣ1 i + hΣ2 i) ∩ B∗ where Σ1 ∪ Σ2 = Σ, Σ1 ∩ Σ2 = ∅, and Ed ∈ Σ1 . Since Ed ⊆ B∗ , we only have to prove that Ed ∩ (hΣ1 i + hΣ2 i) 6= ∅. We now replace the operation in hΣi i by 0 , where √ √ C 0 D := (C + D ) ∩ [−1 − λ/ d, 1 + λ/ d]d . √ √ Then C 0 D ⊆ C D, since [−1 − λ/ d, 1 + λ/ d]d ⊆ B∗ . Denoting the paren0 thesizations with respect to 0 by hΣi i0 , it is √ sufficient√to prove that Ed ∩ (hΣ1 i + hΣ2 i0 ) 6= ∅. Since all Ei as well as [−1 − λ/ d, 1 + λ/ d]d are Cartesian products, we may show this coordinatewise. For C, D ⊆ R, let √ √ C 00 D := (C + D ) ∩ [−1 − λ/ d, 1 + λ/ d], and for any family Σ of subsets of Rd , let πi (Σ) denote the family {πi ( A) : A ∈ Σ} of subsets of R, where πi : Rd → R is the ith coordinate projection. We have to show the following: 1 + λ ∈ hπd (Σ1 )i00 + hπd (Σ2 )i00 , (8.5) and for all i ∈ [d − 1],
[−1, 1] ∩ hπi (Σ)i00 6= ∅,
(8.6)
where the parenthesizations are with respect to 00 . First note the following. Claim 1. Any parenthesization with respect to 00 of one or more sets all equal to [−1, 1] contains [−1, 1]. It follows from Claim 1 and induction that Claim 2. Any parenthesization with respect to 00 of two or more sets √ all but one equal to [−1, 1], and the remaining set equal to {1 + λ}, contains [λ, 1 + λ/ d]. √ √ √ The interval [λ, 1 + λ/ d] is non-empty since λ ≤ d/( d − 1) by hypothesis. Since πd (Σ1 ) consists of [−1, 1]’s (perhaps √ none) and a {1 + λ}, we obtain from 00 Claim 2 that either hπd (Σ1 )i ⊇ [λ, 1 + λ/ d] or hπd (Σ1 )i00 = {1 + λ}. Similarly, since πd (Σ2 ) consists only of [−1, 1]’s, we have hπd (Σ2 )i00 ⊇ [−1, 1] by Claim 1, and (8.5) follows.
80
8.3 Perturbations of `1d Let i ∈ [d − 1]. Then πi (Σ) consists of one {1 + λ}, one {−1 − λ}, and 2d − 3 [−1, 1]’s. The parenthesization hπi (Σ)i00 has a unique subexpression hπi (Σ0 )i00 = hπi (Σ1 )i00 00 hπi (Σ2 )i00 such that {1 + √λ} ∈ πi (Σ1 ) and {−1 − λ} ∈ πi (Σ2 ). By 00 Claim 2, either hπi (Σ1 )i√ ⊇ [λ, 1 + λ/ d] or hπi (Σ1 )i00 = {1 + λ}. Similarly, either hπi (Σ2 )i00 ⊇ [−1 − λ/ d, −λ] or hπi (Σ2 )i00 = {−1 − λ}. This gives four cases, namely √ √ • hπi (Σ0 )i00 ⊇ [−1 + λ − λ/ d, 1 − λ + λ/ d], √ • hπi (Σ0 )i00 ⊇ [−1, −λ + λ/ d], √ • hπi (Σ0 )i00 ⊇ [λ − λ/ d, 1], • hπi (Σ0 )i00 ⊇ {0}. 00 In all four cases √ it follows that hπi (Σ0 )i ∩ [−1, 1] 6= ∅ (the first three cases because λ ≤ 1/(1 − / d)), and we obtain (8.6) from the following claim, which is proved similarly to Claim 2.
Claim 3. Any parenthesization with respect to 00 of two or more sets all but one equal to [−1, 1], and the remaining set having non-empty intersection with [−1, 1], has non-empty intersection with [−1, 1]. This finishes the proof.
81
8 Applications to specific high-dimensional spaces
82
Chapter
9
Using illumination and covering of convex bodies In this chapter we use Bezdek’s quantitative illumination parameter to give an asymptotic upper bound for τ (Md ) that depends only on the dimension, and come close to settling the conjectures of Morgan and Cieslik (Conjectures 4.5 and 4.3).
9.1 Illumination Let K be a convex body in a finite-dimensional vector space V. A point p ∈ / K − → illuminates a point q on the boundary of K if the ray pq intersects the interior of K. A set of points P ⊆ Rd \ K illuminates K if each boundary point of K is illuminated by some point in P. Let L(K ) be the smallest size of a set that illuminates K. Note that d ) = 2d . The well-known illumination problem for the d-dimensional cube, L( B∞ d is to show that L(K ) ≤ 2 for all d-dimensional convex bodies K. For large d the best known upper bounds are L(K ) ≤ (2d d ) d (log d + log log d + 5), and when K is centrally symmetric, L(K ) ≤ 2d d(log d + log log d + 5), due to Rogers [48, p. 284]; see also [89]. There are other equivalent formulations of this illumination problem. For example, let L0 (K ) be the smallest number of positive homothets of K, with each homothety ratio less than 1, whose union contains K. Then L(K ) = L0 (K ). See [102] and [70] for surveys on this problem, its history and related problems.
9.2 Quantitative illumination and Steiner minimal trees We consider a quantitative version of the illumination problem. The first was introduced by K. Bezdek [7]. Let K be a centered convex body defining the norm k·kK . Define the illumination parameter of K to be ( ) I (K ) := inf
∑k pi kK : { pi } illuminates K
.
i
83
9 Using illumination and covering of convex bodies
v B u
u − εp
kεpk
` p0
o
p
Figure 9.1 This ensures that far-away light sources are penalized. Bezdek asked for a sharp upper bound of I (K ) in terms of d, and in particular, if such an upper bound exists for d ≥ 3. For d = 2 he showed that I (K ) ≤ 6 with equality for the regular hexagon. d ) = 2d and for the cross Note that I (K ) ≥ L(K ). It is also easily seen that I ( B∞ d polytope I ( B1 ) = 2d. However, it is not intuitively clear that I (K ) can be bounded from above independent of K when the dimension is fixed. In the next section we will find an upper bound. We first show the connection with Steiner minimal trees. Theorem 9.1. For any finite-dimensional normed space M with unit ball B, σ (M) ≤ τ (M) ≤ I ( B ). The fundamental geometric fact needed for the proof is supplied by the following lemma. Lemma 9.2. If p illuminates the boundary point u of the unit ball B, then for all sufficiently small ε > 0, ku − εpk < 1 − ε. Proof. The lemma is trivial if p = λu for some λ. Therefore, assume that p and u are linearly independent and consider the two-dimensional subspace spanned by them (Figure 9.1). Since p illuminates u, we may choose ε 0 > 0 such that the line through o and u − ε 0 p intersects the line ` through u and p in the interior of B. Then clearly for all ε > 0 with ε < ε 0 the line through o and u − εp still intersects ` in the interior of B. Let v = (ku − εpkK )−1 (u − εp). Then the lines vu and op intersect in p0 , say, with k p0 k < k pk. Using similar triangles, ku − εpk = 1 − kεpk/k p0 k < 1 − ε. Proof of Theorem 9.1. Consider a terminal configuration {oui : i ∈ [τ (M)]} such that kui k = 1 for each i ∈ [τ (M)] (cf. Proposition 4.1). Thus the star T joining o to each ui is a K-SMT of {o, u1 , u2 , . . . , uτ (M) }. Let { p1 , . . . , pk } illuminate K. For each j ∈ [k ], let Uj = {ui : p j illuminates ui }. Then {ui } = j Uj . We now estimate the number of points |Uj | in each Uj . By Lemma 9.2 we may find ε > 0 such that kui − εp j kK < 1 − ε for all ui ∈ Uj . S
84
9.3 Quantitative covering Consider the tree T 0 obtained from the star T by replacing, for each ui ∈ Uj , the edge from o to ui by the edge from εp j to ui , and joining the new Steiner point εp j to o. Then T 0 is not shorter than T. This implies that
|U j | =
∑
kui k ≤ kεp j k +
u i ∈U j
∑
kui − εp j k
u i ∈U j
< εk p j k + (1 − ε)|Uj |, and |Uj | < k p j k. Hence τ (M) ≤ ∑kj=1 |Uj | < ∑kj=1 k p j k. Taking the infimum over all sets { pi } that illuminate B, we obtain that τ (M) ≤ I ( B).
9.3 Quantitative covering We now introduce a quantitative covering parameter for any convex body K, and show that in the centrally symmetric case it can be used to bound I (K ) from above. ( ) C (K ) := inf
∑ (1 − λ i ) −1 : K ⊆
[
i
(λi K + ti ), 0 < λi < 1, ti ∈ Rd
.
i
In this way homothets almost as large as K are penalized. Proposition 9.3. For any centered convex body K we have I (K ) ≤ 2C (K ). Proof. Let {λi K + ti } be a finite covering of K, with 0 < λi < 1 for all i. Let ε > 0 be sufficiently small such that all λi + ε < 1. If a boundary point q of K is covered by λi K + ti , then 1 − λi ≤ kti kK ≤ 1 + λi < 2, and the centre of the homothety mapping K to (λi + ε)K + ti , namely pi := (1 − λi − ε)−1 ti , is outside K and illuminates q. Therefore, the set { pi } illuminates K, and ∑i k pi kK < ∑i 2/(1 − λi − ε). Since ε > 0 can be made arbitrarily small, ∑i k pi kK ≤ 2 ∑i (1 − λi )−1 . d ) = 2d+1 . As with I ( K ), it is not clear whether C ( K ) can be Note that C ( B∞ bounded above only in terms of d. Levi [66] showed that any planar convex body can be covered with 7 homothets, each with homothety ratio 1/2; hence C (K ) ≤ 14 for all 2-dimensional convex bodies K. Lassak’s result [61] that√any planar convex body can be covered √ with 4 homothets, each with ratio 1/ 2, improves the3 upper bound to 8 + 4 2. Lassak [62] also showed that any convex body in R can be covered with 28 homothets, each with ratio 7/8; hence C (K ) ≤ 224 for all 3-dimensional convex bodies K. We now show that C (K ) can be bounded from above in terms of its covering density.
Theorem 9.4. For any d-dimensional convex body K, C ( K ) < ( d + 1) e
vol(K − K ) ϑ ( K ). vol(K )
Proof. It is known [89] that for any 0 < λ < 1 there exists a covering of K by homothets {λK + ti : i ∈ [ N ]}, with N≤
vol(K − λK ) vol(K − K ) ϑ ( K ) < λ−d ϑ ( K ). vol(λK ) vol(K )
85
9 Using illumination and covering of convex bodies Choosing λ = d/(d + 1) we obtain N
∑ (1 − λ )
i =1
−1
1 < ( d + 1) 1 + d
< ( d + 1) e
d
vol(K − K ) ϑ(K ) vol(K )
vol(K − K ) ϑ ( K ). vol(K )
Corollary 9.5. For any d-dimensional convex body K, 2d 2 d log d, C (K ) < (e + o (1)) d and if K is centrally symmetric, then C (K ) < (e + o (1))2d d2 log d, Proof. By the Rogers-Shephard inequality (Section 2.3), vol(K − K )/ vol(K ) ≤ (2d d ). Also, ϑ (K ) ≤ d(log d + log log d + 5) for d ≥ 2 (Rogers [88]). By Proposition 9.3 the following is obtained. Corollary 9.6. For any d-dimensional normed space Md with unit ball B, I ( B) < 2d+1 (d + 1)eϑ ( B) < (2e + o (1))2d d2 log d. Corollary 9.7. For any d-dimensional Minkowski space Md , σ(Md ) ≤ τ (Md ) < (2e + o (1))2d d2 log d. In fact we actually have that σ(Md ) ≤ τ (Md ) < 2d+1 (d + 1)eϑ ( B), which doesn’t really help much, since ϑ ( B) < (1 + o (1))d log d by Rogers’ upper bound, which is small compared to 2d . Conjecture 9.8. There exists c > 0 such that for any d-dimensional convex body K, C (K ) < c2d . We also state the following weaker conjecture, which would still be relevant for Steiner minimal trees. Conjecture 9.9. There exists c > 0 such that for any d-dimensional centered convex body B, I ( B) < c2d .
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Chapter
10
k-Steiner minimal trees In this chapter we consider a modification of Steiner minimal trees, where the number of Steiner points in the Steiner tree is bounded above by a natural number k. Such a shortest tree is called a k-Steiner minimal tree (k-SMT) of the given set of terminals. The local k-Steiner problem can then be considered to be the problem of describing all stars which are k-SMTs. This problem is much more difficult than the local Steiner problem. As an indication of the difficulties, note that, in contrast to the case of Steiner minimal trees, the lengths of the edges of the star are important: if {ox1 , . . . , oxk } is a k-SMT of { x1 , . . . , xk }, and yi = λi xi for λi > 0, then it does not necessarily follow that {oy1 , . . . , oyk } is a k-SMT of {y1 , . . . , yk }. An example will be described below that shows that a star 1-SMT can indeed be destroyed by changing the lengths of the edges. As another example of the difficulties that occur, even though it is clear that k-SMTs are also k0 -SMTs for k0 < k, we have no example of a k0 -SMT that is not a k-SMT, where k0 < k. There does not seem to be much hope to find a general characterization of the local structure of k-SMTs (as we did for SMTs in Chapter 7). Even the problem of giving a concrete characterization in dimension 2 (as done for SMTS in Chapter 5) seems to be very difficult. Our goal in this chapter is more modest. We first find a lower bound for σk (Md ) that is exponential in d, independent of the specific space. This lower bound uses the existence of large sets of unit vectors that are well separated, due to Arias-de-Reyna, Ball, and Villa [3]. Then we find an upper bound for the maximum degree σ1 (M2 ) of a Steiner point in a 1-SMT in a two-dimensional Minkowski space M2 . This comes from the paper [98]. We show that σ1 (M2 ) ≤ 6, with equality if, and only, M2 is isometric to the 3-geometry (i.e., with unit ball an affine regular hexagon). Our main tool for this result will be a classification of all Hadwiger configurations of size 6 of a planar convex body.
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10 k-Steiner minimal trees
10.1 General considerations We say that two convex bodies overlap if their interiors intersect, and touch if they intersect but do not overlap. A Hadwiger configuration of a convex body C is a set of mutually non-overlapping translates {C + xi : i = 1, 2, . . . , n} of C such that each C + xi touches C. The maximum size of a Hadwiger configuration of C is called the Hadwiger number H (C ) of C; see Zong’s survey [111] for an overview and further references. It is known that H (C ) ≤ 3d − 1 [49] with equality if, and only, C is affinely equivalent to the d-cube [45]. If C ⊂ R2 , then H (C ) = 6 if C is not a parallelogram [47]. It is known that the Hadwiger number H ( B) of the unit ball B of a Minkowski space M is an upper bound for the maximum degree of a minimum spanning tree of a set of terminals in M [21, 24, 85]. Thus in particular, σ(M) ≤ τ (M) ≤ 3d − 1, which is improved by Corollary 9.7 in the previous chapter. A k-Steiner tree of a finite set of terminals N in a Minkowski space is a Steiner tree of N, with at most k Steiner points. (It is possible for a k-Steiner tree to have less than k Steiner points.) A k-Steiner minimum tree (k-SMT) of N is a shortest k-Steiner tree of N. Let σk (M) [resp. τk (M)] be the maximum degree that a Steiner point [resp. terminal] can have in a k-SMT in the Minkowski space M. It was shown by Cieslik that τk (M) = H ( B), the Hadwiger number of the unit ball B [24]. In particular, if the unit ball B of M2 is not a parallelogram, σ1 (M2 ) ≤ τ1 (M) = 6. It is clear that σ(M) ≤ · · · ≤ σk+1 (M) ≤ σk (M) ≤ · · · ≤ σ1 (M), but unknown whether the strict inequality σk+1 (M) < σk (M) can occur. Note the lower bound σk (M) ≥ σ(M) ≥ 3 when dim M ≥ 2 [26]. We now show that σk (Md ) grows exponentially for fixed k. Theorem 10.1. There exists c > 0 such that for any k ∈ N and d-dimensional Minkowski space Md , ! d/4 c 4 σk (Md ) ≥ . kd 3 A better exponential lower bound was shown for σ1 (`2d ) by Colthurst et al. [29]. We need the following lemma. √ Lemma 10.2 (Arias-de-Reyna, Ball, and Villa [3]). Let 0 < δ < 2. When two points x and y are chosen randomly and independently from the unit ball B of a d-dimensional Minkowski space Md , then the probability that k x − yk ≤ δ is at most !d √ δ 4 − δ2 . 2 Here the probability distribution is taken to be Lebesgue measure normalized on B. The following is a standard consequence [3].
88
10.1 General considerations Corollary 10.3. √There exists c > 0 such that for any d-dimensional Minkowski space Md and any δ ∈ (1, 2) there exists a set U of at least d/2 2 √ c δ 4 − δ2 unit vectors such that the distance between any two is more than δ. Proof. Choose m points randomly and independently. The probability that some two of them are at distance ≤ δ is at most !d √ m δ 4 − δ2 . (10.1) 2 2 d/2 √ If m = ( 2 + o (1)) √ 2 2 , then (10.1) will be less than 1, which implies that δ 4− δ there exists a subset V of the unit ball of at least m points such that any two is at distance > δ. Since δ > 1, it follows from Lemma 2.5 that any two points in b := { xb : x ∈ V, x 6= o} are also at distance > δ. V Bourgain (cf. [41]) proved the above corollary without an explicit value for the base of the exponent, by using Milman’s Quotient-Subspace theorem. It is now clear that Theorem 10.1 will follow from Corollary 10.3 and the following theorem. Theorem 10.4. Let δ > 1. If M contains a set of n unit vectors such that any two are at distance at least δ, then 2( δ − 1) σk (M) ≥ n. kδ 2( δ −1)
Proof. If δ ≤ n/(n − 1), then kδ n ≤ 2, and the theorem becomes trivial. Thus we may assume without loss of generality that δ > n/(n − 1). Let N be a set of n unit vectors such that any two are separated by at least δ. We show that any k-SMT 2( δ −1) of N has a Steiner point of degree at least kδ m. By joining the origin o to each point in N we obtain a 1-Steiner tree, hence a k-Steiner tree, of N of length n. It follows that all k-SMTs of N have length at most n. A tree with vertex set N (i.e., without any Steiner point) has length at least (n − 1)δ. However, (n − 1)δ > n by assumption. Therefore, any k-SMT must have at least one Steiner point. Let T be a k-SMT with vertex set V such that N ⊂ V. If the subgraph of T induced by the set of Steiner points S := V \ V is not connected, then T is a union of a k1 -SMT and a k2 -SMT, with k = k1 + k2 . Since k i ≤ k, it is then sufficient to prove the theorem for the case where the subgraph induced by the Steiner points is connected, which we now assume. We may also assume that T has exactly k Steiner points, of degrees d1 , . . . , dk , say. Let N 0 be the number of points in N that are adjacent to at least one of the Steiner points, and | N 0 | := n0 . Then the subgraph of T induced by N 0 ∪ S is connected, and therefore a tree T 0 . Since the subgraph induced by the Steiner points is connected, each point in N 0 can be adjacent in T 0 to only one Steiner point. Therefore, n 0 + d1 + d2 + · · · + d k = 2( n 0 + k − 1),
89
10 k-Steiner minimal trees and
k
n0 = 2k − 2 + ∑ di .
(10.2)
i =1
Double each edge of T 0 to obtain a multigraph of length 2`( T 0 ), which has an Euler tour. Use the triangle inequality to change this Euler tour into a cycle that visits each point in N 0 , of length at most 2`( T 0 ). Since this cycle also has length at least n0 δ, 1 (10.3) `( T 0 ) ≥ δn0 . 2 There are n + k0 − 1 edges in T and n0 + k − 1 edges in T 0 . Therefore, T has n − n0 edges not in T 0 . Since these edges are between points of N, each has length at least δ. It follows (using (10.3)) that 1 1 `( T ) ≥ `( T 0 ) + (n − n0 − k)δ ≥ δn0 + (n − n0 )δ = (n − n0 )δ. 2 2 Since `( T ) ≤ n, 2( δ ) n. δ Combined with (10.2), this gives that the average degree of a Steiner point is n0 ≥
1 k 2 2( δ − 1) n+2− . di ≥ k i∑ δk k =1 Since the maximum degree of a Steiner point is at least the average degree, the theorem follows.
10.2 1-Steiner minimal trees in Minkowski planes From now on, we consider only 1-Steiner minimal trees in 2-dimensional Minkowski spaces M. It is possible for σ1 (M) to be strictly smaller than τ1 (M), which equals the Hadwiger number of the unit ball, as the following two examples show. In the Euclidean plane, σ1 (M) = 4 [29, 90], while H ( B22 ) = 6. In the rectilinear plane, σ1 (`21 ) = 5, while the Hadwiger number of the square H ( B12 ) = 8. A result of Cieslik [24] gives the upper bound σ1 (`21 ) ≤ 5 (which also follows from Theorem 10.5 below), while the construction in Figure 10.1 gives a 1-SMT with a Steiner point of degree 5. The above-mentioned bound σ1 (M) ≤ 6 is attained by the 3-geometry, as the following example shows. Let the unit ball of the 3-geometry have vertices {v1 , v2 , . . . , v6 }, in this order. Let N = {2v1 , v2 , 2v3 , v4 , 2v5 , v6 }. Then, as can easily be checked, the tree connecting each point of N to o is a 1-SMT of V in the 3-geometry. See Figure 10.2. Incidentally, note that the star connecting each vi to o is not a 1-SMT of N, since there is a spanning tree of length 5. This shows that the lengths of the edges play a role in k-SMTs. As the following theorems states, the 3-geometry is essentially the only Minkowski plane where a Steiner point of degree 6 can occur. The only planes we know
90
10.2 1-Steiner minimal trees in Minkowski planes
s s
P s
s
s
s
Figure 10.1: A 1-SMT in `21
s 2v3 A A A sv2 A A S A o v4 s As A A As v6 s
s2v1
2v5
Figure 10.2: A 1-SMT in the 3-geometry
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10 k-Steiner minimal trees
Figure 10.3: A Type O configuration and its central symmetrization of for which σ1 (M2 ) ≥ 5 are those isometric to `21 or the 3-geometry. Perhaps σ1 (M2 ) ≤ 4 for all other planes. The only plane for which this is known to be true, is the Euclidean plane. Theorem 10.5. For any Minkowski plane M2 , σ1 (M2 ) ≤ 5, except if M2 is isometric to the 3-geometry. It is also not known whether all Minkowski planes admit 1-SMTs with Steiner points of degree at least four. Although this is most likely true, the best known lower bound is σ1 (M2 ) ≥ σ(M2 ) ≥ 3 [26]. Before proving Theorem 10.5 in Section 10.4, we first have to study Hadwiger configurations of convex discs in the plane. In the next section we show that if six translates of a convex disc C all touch C, and no two of the translates have interior points in common, then there are never more than two gaps, i.e., consecutive nontouching pairs of translates. We also characterize the configurations where there are two, one or no gaps. This result is then used to prove Theorem 10.5.
10.3 Hadwiger configurations in the plane Note that a planar Hadwiger configuration has a natural circular order. We say that two consecutive translates in a planar Hadwiger configuration form a gap if they do not touch. The way in which the existence of a Hadwiger configuration {C + xi : i = 1, 2, . . . , 6} is usually proved, is to find an affine regular hexagon with vertex set { x1 , . . . , x6 } inscribed in the unit circle. In this construction, any two consecutive translates touch (see Figure 10.3), i.e., there are no gaps between consecutive translates. In fact, gaps are impossible in any Hadwiger configuration of six translates if, e.g., C is strictly convex. If, however, C has certain straight line segments on its boundary, then it is possible to slide around the translates in the Hadwiger configuration so as to create gaps, as in Figures 10.5 and 10.6. Theorem 10.6, formulated below at the end of Section 10.3.1, will show that there can never be more than two gaps; if two gaps occur then they must be either opposite (as in Figure 10.5) or consecutive (as in Figure 10.6); and the only possible Hadwiger configurations of six translates are as in Figures 10.3, 10.5 and 10.6. We make this discussion precise in Section 10.3.1, where we define so-called Hadwiger
92
10.3 Hadwiger configurations in the plane configurations of type O, I and II. In Section 10.3.2 we prove Theorem 10.6 via Theorem 10.10, which is a Minkowski distance formulation of Theorem 10.6.
10.3.1 Types of Hadwiger configurations W define three types of Hadwiger configurations of six translates of C. We first define two types of convex disc; refer to Figure 10.4. A convex disc C is of type I if there are points a1 , a2 , . . . , a6 in this order on bd C such that • a1 a2 and a4 , a5 are segments contained in bd C, both parallel to a3 a6 , • k a1 − a2 k + k a4 + a5 k > k a3 + a6 k (since these distances are all parallel, the choice of norm does not matter), and • no segment inside C parallel to a1 a2 is longer than a3 a6 . A convex disc C is of type II if there are points a1 , a2 , . . . , a6 in this order in bd C such that • a1 a2 and a4 a5 are parallel segments in bd C, • a2 a3 and a5 a6 are parallel segments in bd C, • the line through a1 parallel to a2 a3 intersects the relative interior of a4 a5 (in p), and • the line through a3 parallel to a1 a2 intersects the relative interior of a5 a6 (in q). Note that a disc of type II is also of type I in two ways: the vector parallel to a1 a2 which translates aff{ a2 , a3 } to aff{ a5 , a6 } is longer than k a1 − a2 k + k a4 − a5 k, and the vector parallel to a2 a3 which translates aff{ a1 , a2 } to aff{ a4 , a5 } is longer than k a2 − a3 kk a5 − a6 k. A Hadwiger configuration with translation vectors { x1 , x2 , . . . , x6 } is of r
a1
Type I r
a6 r a5
r
a2
r
Type II a3 a4r
r
r
1 a6 a@
rq r a5 @r p@ @r a 4
@ @ @
a@ 2 r
a3 r
Figure 10.4: Two special types of convex discs
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10 k-Steiner minimal trees
Figure 10.5: Type I configuration @ @ @ @ @ @ @
@ @ @
@ @ @ @
@ @ @ @ @ @ @
@ @ @ @ @ @ @
Figure 10.6: Type II configuration type O if C is an arbitrary convex disc, and the translation vectors form the vertex set of an affine regular hexagon, i.e.,
{ x1 , x2 , . . . , x6 } = {± x, ±y, ±( x + y)} for some linearly independent x, y (Figure 10.3), type I if C is of type I, and the translation vectors are
{ x1 , x2 , . . . , x6 } = { a3 − a6 , a6 − a3 , a2 − a5 + ε 1 ( a1 − a2 ), a1 − a4 + ε 2 ( a2 − a1 ), a5 − a2 + ε 3 ( a2 − a1 ), a4 − a1 + ε 4 ( a1 − a2 )} for some small ε 1 , ε 2 , ε 3 , ε 4 ≥ 0, where a1 , a2 , . . . , a6 are as in Figure 10.4 (Figure 10.5), type II if C is of type II, and the translation vectors are
{ x1 , x2 , . . . , x6 } = { a3 − q, a2 − a5 , a1 − p, q − a3 + ε 1 ( a2 − a3 ), a5 − a2 + ε 2 ( a2 − a3 ), p − a1 + ε 3 ( a4 − a5 )} for some ε 1 , ε 2 , ε 3 such that ε 1 ≥ ε 2 ≥ 0 and ε 1 , ε 3 > 0, where a1 , a2 , . . . , a6 , p, q are as in Figure 10.4 (Figure 10.6).
94
10.3 Hadwiger configurations in the plane Note that there are at most two gaps in Hadwiger configuration of types I and II. If two gaps occur in a configuration of type I [resp. type II], they are between opposite pairs [resp. consecutive pairs] of translates. Note that there is an overlap between types O and I. Theorem 10.6. A Hadwiger configuration of six translates of a convex disc C is of type O, I or II. The proof is in Section 10.7.
10.3.2 Symmetrization and Hadwiger sets We say that a finite set H of points in a Minkowski plane M2 = (R2 , B) is a Hadwiger set if H ⊂ bd B (i.e., each point has norm 1), and any two points are at a distance of at least 1. Two consecutive points in a Hadwiger set at a distance greater than 1 form a gap. We now summarize the usual technique of passing to centrally symmetric Hadwiger configurations and to Hadwiger sets. For any planar convex disc C, let C 0 := 1/2(C − C ) be its central symmetral, and k·kC0 the norm with unit ball C 0 . We omit the proof of the following well known lemma. Lemma 10.7. Let C be a convex disc in the plane with central symmetral C 0 . The following statements are equivalent: • The translates C + x and C + y of C touch [overlap], • the translates C 0 + x and C 0 + y of the central symmetrization touch [resp. overlap], • k x − ykC0 = 2 [resp. < 2]. Consequently, the following statements are equivalent: • {C + xi : i = 1, 2, . . . , n} is a Hadwiger configuration of C, • {C 0 + xi : i = 1, 2, . . . , n} is a Hadwiger configuration of C 0 , • {(1/2) x1 , (1/2) x2 , . . . , (1/2) xn } is a Hadwiger set in (R2 , C 0 ). By Lemma 10.7 it is sufficient to prove Theorem 10.6 for centrally symmetric C, once we know the following easily proved lemma. Lemma 10.8. A convex disc C is of type I (type II) if, and only if, C 0 is of type I [resp. type II]. Consequently, a Hadwiger configuration {C + xi } is of type O [resp. type I, type II] if, and only if, {C 0 + xi } is of type O [resp. type I, type II]. From now on we assume that C is a centrally symmetric convex disc defining a Minkowski plane M2 = (R2 , C ) with norm k·k = k·kC , i.e., we suppress the subscript. Whenever we refer to distance or length, we mean distance or length as measured by the norm. We now define Hadwiger sets of type O, I or II. A Hadwiger set H = { x1 , x2 , . . . , x6 } is of
95
10 k-Steiner minimal trees
x3 r x4 r
ar r x3
r x2
r
r x1
r rb
x2
x4 r
r
o x5
r
r x1
o
r
r
x6
x5
Type O
r
x6
Type I x4 r
c r@rx3 @
r x5 @ @ @ r x@ 6
o
@
r
@r x2 = b
r
a r x1 Type II
Figure 10.7: Hadwiger sets of Type O, I and II type O if • C is an arbitrary centrally symmetric convex disc, and • H = {± x, ±y, ±( x + y)} for some unit vectors x and y at distance 1, type I if • • • • •
C contains a long segment ab, x4 = − x1 , aff{ x1 , x4 } and aff{ a, b} are parallel, x2 , x3 ∈ ab, and x5 , x6 ∈ (− a)(−b),
type II if • • • • • • • • •
96
C contains non-parallel long segments ab and bc, x1 ∈ ab, k x1 − bk = 1, x2 = b, x3 = x2 − x1 , x4 ∈ (− x1 )(− a) with x4 6= − x1 , x5 ∈ − ab, x4 x5 ≥ 1, and x6 ∈ (− x3 )(−c) with x6 6= −c.
10.3 Hadwiger configurations in the plane We omit the simple proof of the following lemma. Lemma 10.9. Let M2 = (R2 , C ) be any Minkowski plane. Then • C is of type I if, and only if, C has a long segment, • C is of type II if, and only if, C has two non-parallel long segments joined at a common endpoint. Consequently, a Hadwiger configuration {C + xi } is of type O, [I, II] if, and only if, the corresponding Hadwiger set {(1/2) x1 , (1/2) x2 , . . . , (1/2) x6 } is of type O, [resp. I, II]. It is now clear that the next theorem would imply Theorem 10.6. Theorem 10.10. A Hadwiger set of size six in a Minkowski plane is of type O, I or II. Before proving this theorem, we consider one-sided Hadwiger configurations, as explained in the following lemma. Lemma 10.11. Let { x1 , x2 , x3 , x4 } be a Hadwiger set in (R2 , C ) such that x1 = − x4 , and x2 and x3 are on the same side of aff{ x1 , − x4 } with x2 between x1 and x3 . Then there is at most one gap between the pairs xi and xi+1 , i = 1, 2, 3. Furthermore, one of the following cases occurs:
•1 There are no gaps, and x2 = x1 + x3 , •2 The only gap is between x1 and x2 ; x1 x2 is a long segment on the unit circle parallel to ox3 , •3 The only gap is between x2 and x3 ; x2 x3 is a long segment on the unit circle parallel to ox1 , •4 The only gap is between x3 and x4 ; x3 x4 is a long segment on the unit circle parallel to ox2 . Proof. Suppose that both k x1 − x2 k > 1 and k x3 − x4 k > 1. Let y be a unit vector on the same side of aff{ x1 , x4 } as x2 and x3 , such that k x1 − yk = 1. Then x2 and x3 are strictly between y and y − x1 , since k x1 − yk < k x1 − x2 k and
k x4 − (y − x1 )k = k x1 − yk < k x3 − x4 k, using Lemma 2.3. Again by Lemma 2.3, 1 = ky − (y − x1 )k ≥ k x3 − yk ≥ k x2 − x3 k ≥ 1. Thus, x2 and y are two unit vectors at distance 1 from x3 . By Lemma 2.3 it follows that x2 (y − x3 ) is a long segment on the unit circle, parallel to ox3 , and containing y and y − x3 . Since y and y − x3 are at distance 1, we must have y − x3 = x1 , hence k x3 − x4 k = 1, a contradiction. Assume now without loss of generality that k x1 − x2 k = 1. If x3 = x2 − x1 , we have •1 . If x3 is between x2 − x1 and x2 , then x2 x3 = 1, and, by Lemma 2.3, x3 x4 is a long segment on the unit circle, parallel to ox2 . It follows that we have •4 . If x3 is between x4 and x2 − x1 , then k x3 − x4 k = 1, and, again by Lemma 2.3, x3 x2 is a long segment on the unit circle, parallel to ox1 , which is •3 . Note that •2 would arise in the case k x3 − x4 k = 1.
97
10 k-Steiner minimal trees Proof of Theorem 10.10. Let H = { x1 , x2 , . . . , x6 } be the Hadwiger set, with the points x1 , x2 , . . . , x6 in this order on the unit circle. Suppose there are five points of H in a closed half plane bounded by a line through o, say x1 , . . . , x5 , in this order. Then { x1 , x2 , x3 , x4 , − x1 } is still a Hadwiger set, by Lemma 2.3. We now consider two cases, depending on k x2 − x4 k. If k x2 − x4 k = 1, then k x2 − x3 k = 1 = k x3 − x4 k, and by Lemma 2.3, x3 ( x4 − x2 ) and x3 ( x2 − x4 ) are segments on the unit circle, the first parallel to ox2 and the second parallel to ox4 . Thus, C is a parallelogram with vertices ± x3 , ±( x2 − x4 ). Then x1 = x2 − x4 , x5 = − x1 , and x6 ∈ − x2 x3 ∪ − x3 x4 . Thus, H is of type II. If k x2 − x4 k > 1, then, applying Lemma 10.11 to { x1 , x2 , x4 , − x1 }, we obtain a long segment parallel to ox1 and passing through x2 , x3 , x4 , which must then have length at least 2. By Lemma 2.2, C is necessarily a parallelogram with vertices ± x2 , ± x4 , and x1 = (1/2)( x2 − x4 ) = − x5 . It follows that x6 ∈ − x2 x4 , and H is again of type II. We now assume that there are at most four points of H in any closed half plane bounded by a line through o. We distinguish between two cases: 1. Some two points of H are opposite, 2. No two points of H are opposite. Case 1. Some two points of H are opposite, say x1 = − x4 . We apply Lemma 10.11 to H+ := { x1 , x2 , x3 , x4 } and H− := { x4 , x5 , x6 , x1 }, and find that there are at most two gaps in H. If there are no gaps in H, then Case •1 of Lemma 10.11 must apply to both H+ and H− . It follows that if H is not of type O, then x2 , x3 , − x5 , − x6 are collinear unit vectors, giving a long segment on the unit circle, and H is of type I. If there is exactly one gap in H, then Case •1 of Lemma 10.11 must apply to one of H+ or H− , say to H+ , and then •2 , •3 or •4 to H− . Note that •3 gives that H is of type I. We now show that •2 and •4 both reduce to this case as well. If •2 holds, then x4 x5 is a long segment parallel to ox6 . Thus, x2 ∈ x1 (− x5 ), and it follows that x1 x2 and x4 x5 are parallel. However, x1 x2 and ox3 are parallel. It follows that x3 and x6 are opposite, •1 applies to { x6 , x1 , x2 , x3 }, and •3 applies to { x3 , x4 , x5 , x6 }, giving that H is of type I. The case •4 is similarly reduced. If there are two gaps in H, then •1 applies to neither H+ nor H− . If •2 applies to H+ and •3 to H− , then 1 = k x4 − x5 k = k(− x5 ) − x1 k. Thus, − x5 is in the relative interior of x1 x2 . Then x5 is in the relative interior of x4 (− x2 ), and it follows that x4 x6 is a segment on the unit circle of length k x4 − x5 k + k x5 − x6 k > 2, which contradicts Lemma 2.2. Essentially four possibilities remain: 1. •3 applies to both H+ and H− , giving that H is of type I. 2. •2 applies to both H+ and H− , giving that x1 x2 is paralle to ox3 , and x4 x5 is parallel to ox6 . The long segments x1 x2 and x4 x5 must be parallel, though. It follows that x3 and x6 are opposite, and we obtain that H is of type I.
98
10.3 Hadwiger configurations in the plane 3. •4 applies to both H+ and H− , which similarly gives that H is of type I. 4. •2 applies to H+ and •4 to H− . Then x1 x2 ] and x6 x1 are long segments on the unit circle. It follows that x3 is in the relative interior of x4 (− x6 ), and x5 in the relative interior of x4 (− x2 ). This gives that H is of type II. This finishes Case 1. We remark that in Case 1, we did not obtain a type II set with one gap, since in such a set there are no opposite pairs of points. This remark is used in Case 2. Case 2. No two points of H are opposite. For each i = 1, 2, . . . , 6, let ei = +1 or ei = −1 depending on the orientation of 4 xi oxi+3 . Then ei = −ei+3 . Modulo rotation and reflection of the indices, there are only two patterns: e1 . . . e6 = − + + + − −
or
e1 . . . e6 = − + − + − + .
See Figure 10.8. For both patterns we obtain by Lemma 2.3 that
k(− x1 ) − x5 k = k x1 − (− x5 )k ≥ k x1 − x2 k ≥ 1 and
k(− x1 ) − x3 k ≥ k x4 − x3 k ≥ 1. Thus x4 can be replaced by x40 := − x1 , and Case 1 gives that the Hadwiger set H 0 := { x1 , x2 , x3 , x40 , x5 , x6 } is of type O, I or II. Type O is immediately ruled out. If H 0 is of type I, we get k x3 − x40 k = k x3 − x4 k = 1, and by Lemma 2.3, x4 ( x40 − x3 ) is a segment on the unit circle, parallel to ox3 . However, x40 − x3 ∈ aff{− a, −b}, and it follows that x40 − x3 = −b, and b = x1 + x3 . Then we must have x2 = x1 + x3 . Also, by Lemma 2.3, we obtain 1 = k x3 − x40 k = k(− x3 ) − x1 k = k x6 − x1 k for both patterns, and (−b) x6 is on the unit circle, parallel to ox1 . This shows that H is of type II. If H 0 is of type II, then there must be at least two gaps in H 0 , by the remark at the end of Case 1. Then the only possible pattern is − + + + −−, and the four segments x1 x2 , x1 x6 , x3 x40 , x40 x5 are all on the unit circle. It follows that H is also of type II. r x3 r x2 A A r x4 A r x1 A A A r r x5 A x6
r x3 r x2 A A r x4 A r x1 A A A r x rA 5 x
− + + + −−
− + − + −+
6
Figure 10.8: Two possibilities in Case 2
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10.4 Proving the two-dimensional bound Assume that there exists a 1-SMT in M2 = (R2 , C ) with a Steiner point s of degree d ≥ 6. Our goal is to show that C is necessarily an affine regular hexagon. We may assume that the 1-SMT consists only of s and its neighbors, since a shorter 1-Steiner tree connecting s and its neighbors would shorten the original 1-SMT. Denote the neighbors of s by p1 , p2 , . . . , pd . By scaling and translating, we may assume that s = o, and that k pi k ≥ 1 for all i = 1, 2, . . . , d. Since k pi − p j k ≥ max{k pi k, k p j k} (otherwise there is a shorter tree), we obtain pbi pbj ≥ 1 by Lemma 2.5. Thus, H = { p bi : i = 1, 2, . . . , d} is a Hadwiger set. If d > 6, then by Theorem 10.10, for any subset of H of six points, there is a unit distance between some two of the points, say between p bi and p bj . By Lemma 2.5, either k pi − p j k = k pi k or k pi − p j k = k p j k; hence we may replace opi or op j by pi p j in the 1-SMT to obtain a new 1-SMT in which the Steiner point has degree d − 1. By possibly repeating this argument, we may assume that d = 6. Theorem 10.10 gives that H is of type O, I or II. We now show that type O leads to the case where C is an affine regular hexagon, while types I and II lead to contradictions. We now relabel p bi = xi for each i = 1, . . . , 6 as in Figure 10.7. Type O. Suppose that the segment p b2 p b3 is not on the unit circle. We may assume without loss of generality that k p3 k ≤ k p2 k. Since also k p b2 − p b3 k = 1, Lemma 2.5 gives k p2 − p3 k = k p2 k. As before, k p1 − p2 k ≥ max{k p1 k, k p2 k}. If k p2 k > k p1 k, then Lemma 2.5 gives that the segment pb2 ( pb2 − pb3 ) is on the unit circle, a contradiction. Thus, k p2 k ≤ k p1 k = k p1 − p2 k. The 1-Steiner tree T 0 with edges p1 p2 , p2 p3 , p5 pi (i = 3, 4, 6), cannot be shorter than T: 0 ≤ LT0 − LT =
6
6
i =3
i =3
∑ k pb5 − pi k − ∑ k pi k
6
≤
∑ (k pb5 − pbi k + k pbi − pi k − k pi k)
(10.4)
i =3 6
=
∑ (k pb5 − pbi k − 1) = k pb5 − pb3 k − 2.
i =3
Therefore, k p b5 − pb3 k ≥ 2. Since p b5 p b3 ≤ 2, we obtain that (1/2)( p b2 + p b3 ) = (1/2)( pb3 − pb5 ) is a unit vector, i.e., pb2 pb3 is a segment on the unit circle, which is a contradiction. It follows that p b2 p b3 is on the unit circle. Similarly, p b1 p b2 and p b3 p b4 are segments on the unit circle. It follows that the unit ball is an affine regular hexagon. Type I. We first show that k p1 k ≤ k p2 k. Suppose then that k p1 k > k p2 k. By Lemma 2.5, k p1 k = k p1 − p2 k, and p b1 ( p b1 − p b2 ) is a segment on the unit circle. Thus, p b1 − p b2 = − a. Then, since ab is a long segment and a + p b1 = p b2 ∈ ab, we see that p b2 is in the relative interior of ab. Suppose that p b6 is in the relative interior of − ab. Then, shifting the Steiner point s = o to s0 = ε p b4 for some small ε > 0, we obtain a shorter 1-Steiner tree, since
k pb2 − s0 k = k pb3 − s0 k = k pb5 − s0 k = k pb6 − s0 k = 1 > k pb4 − s0 k.
100
10.4 Proving the two-dimensional bound This contradiction shows that p b6 = − a and p b1 − p b6 = p b2 . Since p b2 is in the relative interior of ab, p b1 p b2 = p b1 ( p b1 − p b6 ) is not a segment on the unit circle. By Lemma 2.5, k p1 k ≤ k p6 k = k p1 − p6 k. The 1-Steiner tree T 0 with edges p1 p2 , p1 p6 , p b4 pi , i = 2, 3, 4, 5 cannot be shorter than T, and a calculation similar to (10.4) gives that k p b4 − p b2 k = 2. It follows that (1/2)( p b1 + p b2 ) = (1/2)( p b2 − p b4 ) is a unit vector, i.e., p b1 p b2 is a segment on the unit circle, which is a contradiction. It follows that
k p1 − p2 k = k p2 k ≥ k p1 k, and, similarly,
k p6 − p1 k = k p6 k ≥ k p1 k, k p3 − p4 k = k p3 k ≥ k p4 k, and
k p5 p4 k = k p5 k ≥ k p4 k. Since k p3 − p4 k = k p3 k, and op4 is parallel to ab, we find p\ 3 − p4 ∈ ab. It follows −→ that the ray p4 b intersects aff{o, p3 } in y, say. By Lemma 2.4,
k p4 − zk + kz − p3 k = k p4 − p3 k for any z ∈ p4 y. By considering the 1-Steiner tree with edges p3 z, p4 z, p1 z, p1 p2 , p4 p5 , p1 p6 , we obtain that k p1 − zk ≥ k p1 − p4 k for all z ∈ p4 y. It follows that the unit ball C has a supporting line at p b4 parallel to ob. Then k a − bk ≤ 1, a contradiction. Type II. If k p1 k > k p6 k, then (since k p b1 − p b6 k = 1) Lemma 2.5 gives that the segment p b1 ( p b1 − p b6 ) is a on the unit circle, a contradiction. Therefore,
k p1 k ≤ k p6 k = k p1 − p6 k. Similarly,
k p3 k ≤ k p4 k = k p3 − p4 k and
k p1 k ≤ k p2 k = k p1 − p2 k. We may therefore form a new 1-SMT by replacing op6 by p1 p6 , op4 by p3 p4 , and op2 by p1 p2 . The Steiner point o is now joined only to p1 , p3 , p5 . Shift the Steiner point to s0 = εp2 for some small ε > 0. This gives a shorter 1-Steiner tree, since
k p5 − s0 k + ks0 − p1 k = k p5 − p1 k = k p5 − ok + ko − p1 k (by Lemma 2.4), and ko − p3 k > ks0 − p3 k. This is a contradiction.
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Bibliography
[1] M. Alfaro, M. Conger, K. Hodges, A. Levy, R. Kochar, L. Kuklinski, Z. Mahmood, and K. von Haam. The structure of singularities in Φ-minimizing networks in R2 . Pacific J. Math., 149:201–210, 1991. [2] I. Anderson. Combinatorics of finite sets. Dover Publications Inc., Mineola, NY, 2002. Corrected reprint of the 1989 edition. [3] J. Arias-de-Reyna, K. Ball, and R. Villa. Concentration of the distance in finitedimensional normed spaces. Mathematika, 45(2):245–252, 1998. [4] S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM, 45:753–782, 1998. [5] S. Arora. Approximation schemes for NP-hard geometric optimization problems: a survey. Math. Program., 97(1-2, Ser. B):43–69, 2003. [6] M.W. Bern and R.L. Graham. The shortest-network problem. Scientific American, January:84–89, 1988. [7] K. Bezdek. Research problem 46. Period. Math. Hungar., 24:119–121, 1992. ¨ [8] A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White, and G.M. Ziegler. Oriented matroids, volume 46 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition, 1999. [9] V. Boltyanski, H. Martini, and V. Soltan. Geometric methods and optimization problems, volume 4 of Combinatorial Optimization. Kluwer Academic Publishers, Dordrecht, 1999. [10] M. Bowron and S. Rabinowitz. Solution to problem 10526. Amer. Math. Monthly, 104:979–980, 1997. [11] M. Brazil. Steiner minimum trees in uniform orientation metrics. In Steiner trees in industry, volume 11 of Comb. Optim., pages 1–27. Kluwer Acad. Publ., Dordrecht, 2001.
103
Bibliography [12] M. Brazil, J. H. Rubinstein, D. A. Thomas, J. F. Weng, and N. C. Wormald. Gradient-constrained minimum networks. I. Fundamentals. J. Global Optim., 21(2):139–155, 2001. [13] M. Brazil, D.A. Thomas, B. K. Nielsen, P. Winter, C. Wulff-Nilsen, and M. Zachariasen. A novel approach to phylogenetic tress: d-dimensional geometric steiner trees. Networks, 53:104–111, 2009. [14] M. Brazil, D. A. Thomas, and J. F. Weng. Minimum networks in uniform orientation metrics. SIAM J. Comput., 30(5):1579–1593 (electronic), 2000. [15] M. Brazil, D. A. Thomas, and J. F. Weng. Gradient-constrained minimum networks. II. Labelled or locally minimal Steiner points. J. Global Optim., 42(1):23–37, 2008. [16] M. Brazil and M. Zachariasen. Steiner trees for fixed orientation metrics. J. Global Optim., 43(1):141–169, 2009. [17] J. Brimberg and R. F. Love. Estimating travel distances by the weighted ` p norm. Nav. Res. Logist., 38(2):241–259, 1991. [18] J. Brimberg and R. F. Love. Global convergence of a generalized iterative procedure for the minisum location problem with l p distances. Oper. Res., 41(6):1153–1163, 1993. [19] G. D. Chakerian and M. A. Ghandehari. The Fermat problem in Minkowski spaces. Geom. Dedicata, 17:227–238, 1985. [20] H. Chen, C.-K. Cheng, A. B. Kahng, I. M˘andoiu, Q. Wang, and B. Yao. The Y-architecture for on-chip interconnect: analysis and methodology. In Proc. Intern. Conf. CAD (ICCAD’03), pages 13–19. ACM, 2003. ¨ [21] D. Cieslik. Knotengrade kurzester B¨aume in endlich-dimensionalen Banachr¨aumen. In Proceedings of the 7th Fischland Colloquium, II (Wustrow, 1988), number 39, pages 89–93, 1990. [22] D. Cieslik. The vertex-degrees of Steiner minimal trees in Minkowski planes. In R. Bodendiek and R. Henn, editors, Topics in combinatorics and graph theory (Oberwolfach, 1990), pages 201–206. Physica-Verlag, Heidelberg, 1990. [23] D. Cieslik. The 1-Steiner-minimal-tree problem in Minkowski-spaces. Optimization, 22(2):291–296, 1991. [24] D. Cieslik. 1-Steiner-minimal-trees in metric spaces. In Proceedings of the Twenty-fourth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1993), volume 98, pages 39–49, 1993. [25] D. Cieslik. Steiner minimal trees, volume 23 of Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Dordrecht, 1998.
104
Bibliography [26] D. Cieslik. Shortest connectivity: An introduction with applications in phylogeny, volume 17 of Combinatorial Optimization. Springer-Verlag, New York, 2005. [27] E. J. Cockayne. On the Steiner problem. Canad. Math. Bull., 10:431–450, 1967. [28] T. Colthurst, C. Cox, J. Foisy, H. Howards, K. Kollett, H. Lowy, and S. Root. Networks minimizing length plus the number of Steiner points. In D.-Z. Du and P. M. Pardalos, editors, Network optimization problems, pages 23–26. World Scientific Publishing, Singapore, 1993. [29] M. Conger. Energy-minimizing networks in Rn . Honours thesis, Williams College, Williamstown MA, 1989. [30] R. Courant and H. Robbins. What is Mathematics? Oxford Univ. Press, Oxford, 1941. [31] A. De Bonis, G.O.H. Katona, and K.J. Swanepoel. Largest family without A ∪ B ⊆ C ∩ D. J. Combin. Theory Ser. A, 111(2):331–336, 2005. [32] Z. Drezner, editor. Facility Location, A Survey of Applications and Methods. Springer-Verlag, New York, 1995. [33] Z. Drezner and G.O. Wesolowsky. Sensitivity analysis to the value of p of the ` p distance Weber problem. Ann. Oper. Res., 111:135–150, 2002. [34] D.-Z. Du, B. Gao, R.L. Graham, Z.C. Liu, and P.J. Wan. Minimum Steiner trees in normed planes. Discrete Comput. Geom., 9(4):351–370, 1993. [35] D.-Z. Du and F.K. Hwang. Reducing the Steiner problem in a normed space. SIAM J. Comput., 21(6):1001–1007, 1992. [36] R. Durier. The Fermat-Weber problem and inner product spaces. J. Approx. Theory, 78(2):161–173, 1994. [37] R. Durier and C. Michelot. Geometrical properties of the Fermat-Weber problem. Europ. J. Oper. Res., 20:332–343, 1985. ¨ [38] P. Erdos. On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc., 51:898–902, 1945. ¨ [39] U. Foßmeier and M. Kaufmann. Solving rectilinear Steiner tree problems exactly in theory and practice. In G. Woeginger, editor, Proceedings of the fifth annual European symposium on algorithms, volume 1284 of Lecture Notes in Computer Science, pages 171–185. Springer-Verlag, 1997. ¨ [40] Z. Furedi and P.A. Loeb. On the best constant for the Besicovitch covering theorem. Proc. Amer. Math. Soc., 121(4):1063–1073, 1994. [41] M.R. Garey, R.L. Graham, and D.S. Johnson. The complexity of computing Steiner minimal trees. SIAM J. Appl. Math., 32:835–839, 1977.
105
Bibliography [42] M.R. Garey and D.S. Johnson. The rectilinear Steiner tree problem is NPcomplete. SIAM J. Appl. Math., 32:826–834, 1977. [43] E.N. Gilbert and H.O. Pollak. Steiner minimal trees. SIAM J. Appl. Math., 16:1–29, 1968. ¨ die Anzahl der konvexen Korper, ¨ [44] H. Groemer. Absch¨atzungen fur die einen ¨ ¨ konvexen Korper beruhren. Monatsh. Math., 65:74–81, 1961. [45] P.M. Gruber. Convex and discrete geometry, volume 336 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, 2007. ¨ [46] B. Grunbaum. On a conjecture of H. Hadwiger. Pacific J. Math., 11:215–219, 1961. ¨ [47] B. Grunbaum. Borsuk’s problem and related questions. In Proc. Sympos. Pure Math., Vol. VII, pages 271–284. Amer. Math. Soc., Providence, R.I., 1963. ¨ ¨ Treffanzahlen bei translationsgleichen Eikorpern. Archiv [48] H. Hadwiger. Uber Math., 8:212–213, 1957. [49] M. Hanan. On Steiner’s problem with rectilinear distance. SIAM J. Appl. Math., 14:255–265, 1966. [50] O. Hanner. Intersections of translates of convex bodies. Math. Scand., 4:65–87, 1956. [51] A.B. Hansen. On a certain class of polytopes associated with independence systems. Math. Scand., 41(2):225–241, 1977. ˚ Lima. The structure of finite-dimensional Banach spaces [52] A.B. Hansen and A. with the 3.2. intersection property. Acta Math., 146(1-2):1–23, 1981. [53] F.K. Hwang, D.S. Richards, and P. Winter. The Steiner tree problem, volume 53 of Ann. Discrete Math. North-Holland, Amsterdam, 1992. [54] A.O. Ivanov and A.A. Tuzhilin. Minimal networks. CRC Press, Boca Raton, FL, 1994. The Steiner problem and its generalizations. ¨ [55] V. Jarn´ık and M. Kossler. O minimalnich grafeth obeahujicich n danijch bodu. Cas. Pest. Mat. Fys., 63:223–235, 1934. [56] M.I. Kadets and M.G. Snobar. Certain functionals on the minkowski compactum. Mat. Zametki, 10:453–457, 1971. (Russian). English transl. Math. Notes 10 (1971), 694–696. ¨ [57] B. Korte, H.J. Promel, and A. Steger. Steiner trees in VLSI-layout. In B. Korte et al, editor, Paths, Flows, and VLSI-Layout, volume 9 of Algorithms and Combinatorics, pages 185–214. Springer-Verlag, Berlin, 1990.
106
Bibliography [58] B. Korte and J. Neˇsetˇril. Vojtˇech Jarn´ık’s work in combinatorial optimization. Discrete Math., 235(1-3):1–17, 2001. Combinatorics (Prague, 1998). [59] Y.S. Kupitz and H. Martini. Geometric aspects of the generalized FermatTorricelli problem. In Intuitive Geometry, volume 6 of Bolyai Soc. Math. Stud., pages 55–127, 1997. [60] M. Lassak. Covering a plane convex body by four homothetical copies with the smallest positive ratio. Geom. Dedicata, 21(2):157–167, 1986. [61] M. Lassak. Covering a three-dimensional convex body by smaller homothetic copies. Beitr¨age Algebra Geom., 39(2):259–262, 1998. [62] G.R. Lawlor and F. Morgan. Paired calibrations applied to soap films, immiscible fluids, and surfaces and networks minimizing other norms. Pacific J. Math., 166:55–82, 1994. [63] D.T. Lee, C.-F. Shen, and C.-L. Ding. On Steiner tree problem with 45◦ routing. In 1995 International Conference on Circuits and Systems, pages 1680–1683. IEEE, 1995. [64] D.T. Lee and C.-F. Shen. The Steiner minimal tree problem in the λ-geometry plane. In Tetsuo Asano et al., editor, Proceedings of the seventh international symposium on Algorithms and Computation, volume 1178 of Lecture Notes in Computer Science, pages 247–255. Springer-Verlag, 1996. ¨ [65] F.W. Levi. Ein geometrisches Uberdeckungsproblem. Arch. Math. (Basel), 5:476–478, 1954. [66] Y.Y. Li, S.K. Cheung, K.S. Leung, and C.K. Wong. Steiner tree constructions in λ3 metric. IEEE Trans. Circuits Systems II, 45:563–574, 1998. [67] G.H. Lin and G.L. Xue. The Steiner problem in λ4 -geometry plane. In 1998 International Symposium on Algorithms and Computation, volume 1533 of Lecture Notes in Computer Science, pages 327–336. Springer-Verlag, 1998. [68] G.-H. Lin, G.L. Xue, and D. Zhou. Approximating hexagonal Steiner minimal trees by fast optimal layout of minimum spanning trees. In 1999 International Conference on Computer Design, pages 392–398. IEEE, 1999. [69] H. Martini and V. Soltan. Combinatorial problems on the illumination of convex bodies. Aequationes Math., 57(2-3):121–152, 1999. [70] H. Martini and K.J. Swanepoel. The geometry of Minkowski spaces—a survey. II. Expo. Math., 22(2):93–144, 2004. [71] H. Martini and K.J. Swanepoel. Low-degree minimal spanning trees in normed spaces. Appl. Math. Lett., 19(2):122–125, 2006. [72] H. Martini, K.J. Swanepoel, and P.O. de Wet. Absorbing angles, Steiner minimal trees and antipodality. J. Optim. Theory Appl., to appear, 2009.
107
Bibliography [73] H. Martini, K.J. Swanepoel, and G. Weiß. The geometry of Minkowski spaces—a survey. I. Expo. Math., 19(2):97–142, 2001. [74] H. Martini, K. J. Swanepoel, and G. Weiß. The Fermat-Torricelli problem in normed planes and spaces. J. Optim. Theory Appl., 115(2):283–314, 2002. [75] C.M. McGregor. Finite-dimensional normed linear spaces with numerical index 1. J. London Math. Soc. (2), 3:717–721, 1971. [76] Z.A. Melzak. On the problem of Steiner. Canad. Math. Bull., 4:143–148, 1961. [77] K. Menger. Some applications of point-set methods. 32(4):739–760, 1931.
Ann. of Math. (2),
¨ [78] Y. Mimura. Uber die Bogenl¨ange. In K. Menger, editor, Ergebnisse eines Mathematischen Kolloquiums, volume 4, pages 20–22. Teubner, Leipzig und Wien, 1933. [79] F. Morgan. The cone over the Clifford torus in R4 is Φ-minimizing. Math. Ann., 289(2):341–354, 1991. [80] F. Morgan. Minimal surfaces, crystals, shortest networks, and undergraduate research. Math. Intelligencer, 14(3):37–44, 1992. [81] F. Morgan. Riemannian Geometry, A Beginner’s Guide. A. K. Peters, 2nd ed. edition, 1998. [82] F. Morgan. Geometric measure theory. Academic Press Inc., San Diego, CA, third edition, 2000. A beginner’s guide. [83] S. Reisner. Certain Banach spaces associated with graphs and CL-spaces with 1-unconditional bases. J. London Math. Soc. (2), 43(1):137–148, 1991. [84] G. Robins and J.S. Salowe. Low-degree minimum spanning trees. Discrete Comput. Geom., 14:151–165, 1995. [85] R.T. Rockafellar. Convex Analysis. Princeton University Press, Princeton, 1997. [86] O. Rodrigues. Sur le nombre de main`eres d´effectuer un produit de n facteurs. J. Math. Pures Appl., 3((1)):549, 1838. [87] C.A. Rogers and G.C. Shephard. The difference body of a convex body. Arch. Math. (Basel), 8:220–233, 1957. [88] C.A. Rogers and C. Zong. Covering convex bodies by translates of convex bodies. Mathematika, 44(1):215–218, 1997. [89] J.H. Rubinstein, D.A. Thomas, and J.F. Weng. Degree five Steiner points cannot reduce network costs for planar sets. Networks, 22:531–537, 1992. [90] M. Sarrafzadeh and C.K. Wong. Hierarchical Steiner tree construction in uniform orientations. IEEE Trans. Computer-Aided Design, 11:1095–1103, 1992.
108
Bibliography [91] R. Schneider. Convex Bodies: the Brunn-Minkowski theory. Encyclopedia of Mathematics and its Applications 44. Cambridge University Press, 1993. ¨ [92] E. Schroder. Vier combinatorische Probleme. Z. fur ¨ Math. Phys., 15:361–376, 1870. ¨ [93] A. Schurmann and K.J. Swanepoel. Three-dimensional antipodal and normequilateral sets. Pacific J. Math., 228(2):349–370, 2006. [94] R.P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. [95] K.J. Swanepoel. Vertex degrees of Steiner minimal trees in l dp and other smooth Minkowski spaces. Discrete Comput. Geom., 21(3):437–447, 1999. [96] K.J. Swanepoel. Balancing unit vectors. J. Combin. Theory Ser. A, 89(1):105– 112, 2000. [97] K.J. Swanepoel. Gaps in convex disc packings with an application to 1-Steiner minimum trees. Monatsh. Math., 129(3):217–226, 2000. [98] K.J. Swanepoel. The local Steiner problem in normed planes. Networks, 36(2):104–113, 2000. [99] K.J. Swanepoel. Quantitative illumination of convex bodies and vertex degrees of geometric Steiner minimal trees. Mathematika, 52(1-2):47–52, 2005. [100] K.J. Swanepoel. The local Steiner problem in finite-dimensional normed spaces. Discrete Comput. Geom., 37(3):419–442, 2007. ´ Recent results on illumination problems. In Intuitive geometry (Bu[101] L. Szabo. dapest, 1995), volume 6 of Bolyai Soc. Math. Stud., pages 207–221. J´anos Bolyai Math. Soc., Budapest, 1997. [102] H. Tamvakis. Problem 10526. Amer. Math. Monthly, 103:427, 1996. [103] A.C. Thompson. Minkowski Geometry. Encyclopedia of Mathematics and its Applications 63. Cambridge University Press, 1996. [104] D.M. Warme, P. Winter, and M. Zachariasen. Exact Algorithms for Plane Steiner Tree Problems: A Computational Study. In D.-Z. Du, J. M. Smith, and J. H. Rubinstein, editors, Advances in Steiner Trees, pages 81–116. Kluwer Academic Publishers, Boston, 2000. [105] R.E. Wendell and A.P. Hurter, Jr. Location theory, dominance, and convexity. Oper. Res., 21:314–321, 1973. [106] P. Winter and M. Zachariasen. Euclidean Steiner minimum trees: an improved exact algorithm. Networks, 30:149–166, 1997.
109
Bibliography [107] G.Y. Yan, A. Albrecht, G.H.F. Young, and C.K. Wong. The Steiner tree problem in orientation metrics. J. Comput. System Sci., 55(3):529–546, 1997. [108] C. Z˘alinescu. Convex analysis in general vector spaces. World Scientific Publishing Co. Inc., River Edge, NJ, 2002. [109] G.M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. [110] C. Zong. The kissing numbers of convex bodies — a brief survey. Bull. London Math. Soc., 30:1–10, 1998.
110
