On the Volume of the Convex Hull of d + 1 Segments in Rd Gennadiy Averkov and Horst Martini Abstract Let d ≥ 2, m ≥ d, and u1 , . . . , um be non-zero vectors linearly spanning Rd . We consider the class P(u1 , . . . , um ) of convex polytopes P in Rd such that P = conv(I1 ∪. . .∪Im ), where, for j = 1, . . . , m, the set Ij is a translate of the segment conv{o, uj }. By v(u1 , . . . , um ) we denote the minimum among the volumes of all polytopes from P(u1 , . . . , ud ), and by P0 (u1 , . . . , uk ) the subclass of P(u1 , . . . , um ) minimizing the volume. It is known that 1 v(u1 , . . . , ud ) = d! det(u1 , . . . , ud ). In this note we evaluate v(u1 , . . . , ud+1 ) and study the properties of P0 (u1 , . . . , ud+1 ).
2000 Mathematics Subject Classification. Primary: 52B11 Secondary: 52A40 Key words and phrases. Geometric inequality, minimum width, Minkowski space, polytope, reduced body, simplex, volume
1
Introduction
Let d ≥ 2. The origin in Rd is denoted by o. The abbreviations conv and vol stand for convex hull and volume, respectively. We use the notations P(u1 , . . . , um ), P0 (u1 , . . . , um ), and v(u1 , . . . , um ) as they are introduced in the abstract. For the case m = d the class P0 (u1 , . . . , um ) and the corresponding quantity v(u1 , . . . , um ) were studied in [7] and [5]. In particular, in the above papers it was shown that 1 v(u1 , . . . , ud ) = | det(u1 , . . . , ud )|, (1) d! where det(u1 , . . . , ud ) denotes the determinant of the matrix with columns u1 , . . . , ud . We perform an analogous study for the case m = d + 1. Theorem 1. Let d ≥ 2, and u1 , . . . , ud+1 be non-zero vectors linearly spanning Rd . Then the following statements hold true. I. The quantity v(u1 , . . . , ud+1 ) is equal to the maximum of v(u01 , . . . , u0d ) over all possible subsets {u01 , . . . , u0d } of {u1 , . . . , ud+1 }. II. The class P0 (u1 , . . . , ud+1 ) necessarily contains simplices. III. Every polytope from P0 (u1 , . . . , ud+1 ) has at most 2d vertices. ¤ The statement of Theorem 1 can be reformulated in terms of Minkowski geometry (that is, the geometry of finite dimensional normed spaces); see [8] and [6]. In fact, if {±u1 , . . . , ±um } is a set of vertices of some o-symmetric d-dimensional convex polytope B and Md is the normed space whose unit ball is B, then the class P0 (u1 , . . . , um ) is precisely the class of convex bodies whose minimum width, measured with respect to Md , is equal to one and whose volume is minimal. It should also be mentioned that the elements of P(u1 , . . . , um ) contain all Md -reduced bodies, which were introduced in [4]. For more information on minimum width and reduced bodies in Minkowski spaces see [4], [3], and [1]. 1
2
The proof
Proof of Theorem 1. For every subset {u01 , . . . , u0d } of {u1 , . . . , ud+1 } the inequality v(u01 , . . . , u0d ) ≤ v(u1 , . . . , ud+1 } is obvious. We show that for some u01 , . . . , u0d as above the reverse inequality is fulfilled. Let us consider the o-symmetric polytope B := conv{±u1 , . . . , ±ud+1 }. For i = 1, . . . , d + 1 we also define Bi := conv ({±u1 , . . . , ±ud+1 } \ {±ui }) . Without loss of generality we assume that the volume of Bi is maximal for i = d + 1 and define the parallelotope ( d ) X ti ui : i = 1, . . . , d, −1 ≤ ti ≤ 1 . B := i=1
It Pdis well known that B ⊆ B (see, for example, [2], and [6, Section 6]). Consequently, ud+1 = i=1 βi ui for some −1 ≤ βi ≤ 1 with i = 1, . . . , d. We introduce numbers c1 , . . . , cd ∈ {−1, 1} such that sign βi ∈ {0, ci } for every i = 1, . . . , d. Further on, let σ be a permutation over {1, . . . , d} such that |βσ1 | ≤ |βσ2 | ≤ · · · ≤ |βσd |. Then we define the points p1 , p2 , . . . , pd by pi :=
d X
cσj uσj .
j=i
Let us show that the simplex T := conv{o, p1 , . . . , pd } belongs to P0 (u1 , . . . , ud+1 ). We introduce the values α1 , . . . , αd by the formulas α1 := |βσ1 |, αi := |βσi | − |βσi−1 |
(2 ≤ i ≤ d).
It is easy to verify that for every j = 1, . . . , d we have 0 ≤ αj ≤ 1 and Now let us show that ud+1 ∈ T. We have à ! d d d d d X X X X X 1− αi o + αi p i = αi p i = αi cσj uσj i=1
i=1
i=1
=
i=1
X
αi cσj uσj =
=
à j d X X j=1
1≤i≤j≤d d X
j=i
|βσj |cσj uσj =
j=1
d X j=1
Pj
i=1 αi
= |βσj | ≤ 1.
! αi
cσj uσj
i=1
βσj uσj =
d X
βj uj = ud+1 .
j=1
Thus, ud+1 is a convex combination of points o, p1 , . . . , pd , and hence ud+1 ∈ T. Obviously, T can be represented by à ! d [ T = conv conv{o, p1 } ∪ conv{pi−1 , pi } . i=2
But the segement conv{o, p1 } is a translate of conv{o, uσ1 }, while, for every i = 2, . . . , d, the segment conv{pi−1 , pi } is a translate of conv{o, uσi }. Since conv{o, ud+1 } ⊆ T, we see that T ∈ P(u1 , . . . , ud ). By construction, vol(T ) =
1 1 1 | det(p1 , . . . , pd )| = | det(uσ1 , . . . , uσd )| = | det(u1 , . . . , ud )| = v(u1 , . . . , ud ). d! d! d!
Hence, v(u1 , . . . , ud ) ≥ v(u1 , . . . , ud+1 ). This shows Part I and (since T is a simplex) also Part II of the theorem. 2
We prove Part III by contradiction. Assume that P is a polytope from P0 (u1 , . . . , ud+1 ) with at least 2d + 1 vertices. Let J1 , . . . , Jd+1 be segments in Rd such that, for i = 1, . . . , d + 1, the segment Ji is a translate of conv{o, ui } and P = conv(J1 ∪ . . . ∪ Jd+1 ). Since P has at least 2d + 1 vertices, at least one of the endpoints of Jd+1 is also a vertex of P. Hence (1)
vol(P ) > vol(conv(J1 ∪ . . . ∪ Jd )) ≥
1 | det(u1 , . . . , ud )| = vol(T ), d!
and we arrive at the contradiction. This finishes the proof of Part III.
References [1] G. Averkov and H. Martini, On reduced polytopes and antipodality, Adv. Geom., 10pp., to appear, 2008+. [2] C. H. Dowker, On minimum circumscribed polygons, Bull. Amer. Math. Soc. 50 (1944), 120–122. MR 5,153m [3] M. Lassak, Characterizations of reduced polytopes in finite-dimensional normed spaces, Beitr¨age Algebra Geom. 47 (2006), no. 2, 559–566. MR 2007m:52007 [4] M. Lassak and H. Martini, Reduced bodies in Minkowski space, Acta Math. Hungar. 106 (2005), no. 1-2, 17–26. MR 2005m:52008 [5] H. Martini, Das Volumen spezieller konvexer Polytope, Elem. Math. 44 (1989), no. 5, 113– 115. MR 90k:52011 [6] H. Martini, K. J. Swanepoel, and G. Weiß, The geometry of Minkowski spaces – a survey, Part I, Expo. Math. 19 (2001), no. 2, 97–142. MR 1 835 964 [7] P. McMullen, The volume of certain convex sets, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 1, 91–97. MR 83a:52008 [8] A. C. Thompson, Minkowski Geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 97f:52001 G. Averkov Faculty of Mathematics, University of Magdeburg, 39106 Magdeburg, Germany e-mail:
[email protected]
H. Martini Faculty of Mathematics, University of Technology, 09107 Chemnitz, Germany e-mail:
[email protected]
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