On Birkhoff orthogonality and isosceles orthogonality in ... - Springer Link

Aequat. Math. 83 (2012), 153–189 c Springer Basel AG 2011 0001-9054/12/010153-37 published online September 18, 2011 DOI 10.1007/s00010-011-0092-z

Aequationes Mathematicae

On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces Javier Alonso, Horst Martini and Senlin Wu

Abstract. We survey mainly recent results on the two most important orthogonality types in normed linear spaces, namely on Birkhoff orthogonality and on isosceles (or James) orthogonality. We lay special emphasis on their fundamental properties, on their differences and connections, and on geometric results and problems inspired by the respective theoretical framework. At the beginning we also present other interesting types of orthogonality. This survey can also be taken as an update of existing related representations. Mathematics Subject Classification (2000). 46B20, 46C15, 52A21. Keywords. Real normed linear space, Minkowski space, inner product space, Birkhoff orthogonality, isosceles orthogonality, James orthogonality, bisectors, Zindler curves.

Contents 1. 2. 3. 4.

Introduction Notations Some orthogonality types Properties of Birkhoff and isosceles orthogonalities 4.1. Birkhoff orthogonality 4.1.1. Characterizations 4.1.2. Homogeneity 4.1.3. Symmetry 4.1.4. Existence

154 155 156 159 159 159 160 161 162

J. Alonso’s research was supported by MICINN (Spain) and FEDER (UE) grant MTM200805460, and by Junta de Extremadura grant GR10060 (partially financed with FEDER). Senlin Wu’s research was supported by National Natural Science Foundation of China (grant number 11001068), a foundation from the Ministry of Education of Heilongjiang Province (grant number 11541069), a foundation from Harbin University of Science and Technology (grant number 2009YF028), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and by Deutsche Forschungsgemeinschaft.

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J. Alonso, H. Martini and S. Wu

4.1.5. Uniqueness 4.1.6. Additivity 4.1.7. Extension 4.1.8. Orthogonal diagonals and related results 4.2. Isosceles orthogonality 4.2.1. Homogeneity 4.2.2. Additivity 4.2.3. Existence and uniqueness 4.2.4. Extension 4.2.5. Orthogonal diagonals and related results 5. The differences between Birkhoff orthogonality and isosceles orthogonality 5.1. Geometric consequences of the differences 5.1.1. Bisectors in a Minkowski plane 5.1.2. Zindler curves in Minkowski planes 5.2. Quantitative characterizations of the differences 6. Connections between Birkhoff orthogonality and isosceles orthogonality References

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163 164 165 165 167 167 167 168 169 169 171 172 172 175 178 182 184

1. Introduction One of the most important concepts in Euclidean Geometry is that of orthogonality. This concept appears not only in the fourth axiom of Euclidean Geometry, but also in many fundamental theorems such as the Pythagorean theorem. When switching from Euclidean Geometry to the geometry of real normed linear spaces, the only thing missing is the fourth Euclidean axiom, which is about orthogonality and angular measure. Therefore, since 1934 (see [89]) many mathematicians have introduced different generalized orthogonality types for normed linear spaces. In this paper we survey some results concerning Birkhoff orthogonality and isosceles (or James) orthogonality, which are the two most important orthogonality types defined for real normed linear spaces. We shall focus on fundamental properties of Birkhoff orthogonality and isosceles orthogonality, differences and connections between these two orthogonality types, and geometric results and problems closely related to them. Our paper can also be taken as an update of existing surveys and monographs (see [2,4,6,8,14,20]) on this topic. We should also mention that the large variety of results from Elementary Geometry and Foundations of Geometry which particularly refer to Euclidean orthogonality should still be seen as a field of further research by discovering analogous results for normed linear spaces, based on generalized orthogonality types. Already existing results in this direction and related to Birkhoff and isosceles orthogonality mainly refer to the coincidence of the heights of triangles

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and orthocentric point systems (see e.g., [49,72,82,86,100]), special arrangements or positions of circles or convex figures (cf. [10,80]), the symmetry of Birkhoff orthogonality (see [18,76,77]), and bisectors of given segments (cf. [50,51,53]). Certainly, many further theorems reflecting or using Euclidean orthogonality are waiting for their extension to normed linear spaces.

2. Notations Although some of the results in this paper are also true in complex spaces, we will, to simplify our representation, only refer to real spaces. Thus, we denote by X a real (Banach or) normed linear space with norm · and origin o. It is assumed that the dimension of the normed linear spaces X considered in this paper, dim X, is at least two. The subsets SX := {x : x = 1} and BX := {x : x ≤ 1} are called the unit sphere and the unit ball of X, respectively. When X is finite dimensional, we say that X is a Minkowski space. In particular, when X is a Minkowski plane, SX and BX are called the unit circle and the unit disc of X, respectively. By X ∗ we denote the dual space of X. For x, y ∈ X, with x = y, we denote by [x, y] := {λx + (1 − λ)y : λ ∈ [0, 1]} the (non-trivial) segment between x and y (while a trivial segment, with x = y, is just a singleton), by x, y := {λx+(1−λ)y : λ ∈ R} the line passing through x and y, and by [x, y := {(1 − λ)x + λy : λ ∈ [0, +∞)} the ray with starting point x passing through y. The distance between x and y, which is equal to → for xy the length of [x, y] in the norm, is measured by x − y. Also we write − x the orientation from x to y, and x for x (x = o), i.e., for the normalization of x. The distance from a point x to a set K is denoted by d(x, K). We say that a normed linear space X is strictly convex if SX does not contain a non-trivial segment. A normed linear space X is smooth if there exists precisely one supporting hyperplane of BX at each point of SX . A (real) inner product space is a special normed linear space with the additional structure of an inner product (·, ·) which, for any x, y, z ∈ X, satisfies: • • • • •

(x, y) = (y, x); (αx, y) = α(x, y), for any α ∈ R; (x + y, z) = (x, z) + (y, z); (x, x) ≥ 0, with equality if and only if x = o; 2 (x, x) = x .

When we say that a normed linear space is Euclidean, we mean that it is an inner product space. In particular, a two-dimensional (real) inner product space is referred to as the Euclidean plane. There are many different ways to characterize inner product spaces among normed linear spaces (a good monograph about this topic is [14]). One of the best known characterizations was given in 1935 by Jordan and von Neumann [63] with the following condition,

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Figure 1. x ⊥B y, but y ⊥B x

Figure 2. x ⊥I y, but x ⊥I αy called the parallelogram law: 2

2

2

2

x + y + x − y = 2 x + 2 y ,

∀x, y ∈ X.

It is well known that an inner product space is strictly convex and smooth.

3. Some orthogonality types In this section we list some orthogonality types introduced for normed linear spaces. Besides Birkhoff orthogonality and isosceles orthogonality we also present some other orthogonalities closely related to at least one of them. Please refer to [6,8] for the properties of many orthogonalities listed below and the relations between each pair of them.

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The first orthogonality type that was defined for general normed linear spaces is probably the one called Roberts orthogonality [89], introduced by Roberts in 1934. Definition 3.1. [89] In a normed linear space X, a vector x is said to be Roberts orthogonal to a vector y (x ⊥R y) if the equality x + αy = x − αy holds for any real number α. Later, in 1935, Birkhoff introduced Birkhoff orthogonality [26], which was revealed to be the most important orthogonality defined for normed linear spaces. Definition 3.2. [26] In a normed linear space X, a vector x is said to be Birkhoff orthogonal to a vector y (x ⊥B y) if the inequality x + αy ≥ x holds for any real number α (Fig. 1). In normed linear spaces Birkhoff orthogonality is actually equivalent to normality as it was introduced by Carathéodory (cf. [28,87,88]). James was the first who provided a comprehensive study of the properties of Birkhoff orthogonality (see [56,57]). Due to this, Birkhoff orthogonality is also referred to as James orthogonality or Birkhoff–James orthogonality. We underline that, for historical reasons, isosceles orthogonality (as given in Definition 3.3 below) is sometimes referred to as James orthogonality, too. To avoid confusion, in this paper we refer to these orthogonalities simply as Birkhoff orthogonality (Definition 3.2) and isosceles orthogonality (Definition 3.3 below). James also observed that the requirements in the definition of Roberts orthogonality are too strong to yield a “nice” orthogonality type. He provided an example of a Minkowski plane such that x y = 0 whenever x ⊥R y holds; see [55]. Moreover, he proved that if for every x it is possible to find in any two dimensional subspace containing x a vector y such that x ⊥R y, then the space is necessarily an inner product space. Inspired by this, James introduced in 1945 isosceles orthogonality and Pythagorean orthogonality. Definition 3.3. [55] In a normed linear space X, a vector x is said to be isosceles orthogonal to a vector y (x ⊥I y) if x + y = x − y (Fig. 2). Definition 3.4. [55] In a normed linear space X, a vector x is said to be Pythag2 2 2 orean orthogonal to a vector y (x ⊥P y) if x − y = x + y . One can easily check that x ⊥R y implies x ⊥B y and x ⊥I y. In 1957, Singer introduced an orthogonality type which is directly related to isosceles orthogonality. Definition 3.5. [95] In a normed linear space X, a vector x is said to be Singer x + y = x − y. orthogonal to a vector y (x ⊥S y) if either x y = 0 or Clearly, for x, y ∈ X with x y = 0, x ⊥S y holds if and only if x ⊥I y.

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In 1962, Carlsson ([31], see also [20]) introduced Carlsson orthogonality, which is actually a family of orthogonalities containing, in particular, isosceles orthogonality and Pythagorean orthogonality. Definition 3.6. [31] Let X be a normed linear space, and αi , βi , γi , i = 1, 2, . . . , m, be fixed real numbers satisfying m m m αi βi2 = αi γi2 = 0, αi βi γi = 1. i=1

i=1

i=1

Then x ∈ X is said to be Carlsson orthogonal to y ∈ X (x ⊥C y) if m

2

αi βi x + γi y = 0.

i=1

α-orthogonality, which is an extension of isosceles and Pythagorean orthogonality, was introduced by Diminnie, Freese, and Andalafte [40] in 1983. It is also a particular case of Carlsson orthogonality. Boussouis introduced in 1995 a similar, but more general orthogonality, which will be referred to as Boussouis orthogonality. Definition 3.7. [30] In a normed linear space X, a vector x is said to be Boussouis orthogonal to a vector y (x ⊥M y) if 2 α(ω) β(ω)x + γ(ω)y dμ(ω) = 0, Ω

where (Ω, μ) is a positive measure space and α, β, γ are μ-measurable functions from Ω to R such that α(ω) = 0 almost everywhere, αβ 2 and αγ 2 are μ-integrable and α(ω)β 2 (ω) dμ(ω) = α(ω)γ 2 (ω) dμ(ω) = 0, α(ω)β(ω)γ(ω) dμ(ω) = 1. Ω

Ω

Ω

There are many other types of orthogonality in normed linear spaces, for example Diminnie orthogonality [39], area orthogonality [2,9], and height orthogonality [13,11]. Saidi [92] introduced an orthogonality type for a sequence of vectors (see [90,91] for characterizations of this orthogonality). Recently, Martini and Spirova introduced two geometrically defined orthogonality types. The first one is deduced from the Minkowskian version of the “three circles theorem” [72,82], and the second one [73], which is called chordal orthogonality, is introduced via properties of chords of the unit circle. In [46] an abstract orthogonality type is introduced and compared with Birkhoff, Pythagorean, and James orthogonality yielding characterizations of inner product spaces via norm invariance and rotation invariance properties. In [12] an orthogonality type is considered which coincides with Birkhoff, isosceles or Pythagorean orthogonality only for the Euclidean case. Inspired by isosceles

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(resp. Pythagorean) orthogonality, in [41] two new orthogonality types are introduced.

4. Properties of Birkhoff and isosceles orthogonalities If “⊥” denotes the orthogonality relation in an inner product space, then this relation has the well known properties listed below. But if “⊥” denotes a generalized orthogonality type in a normed linear space, then these properties are not always satisfied, and they can affect the geometric structure of the space (See Figs. 1, 2). • Non-degeneracy: λx ⊥ μx if and only if λμx = 0. • Simplification: If x ⊥ y then λx ⊥ λy holds for each number λ ∈ R. ∞ • Continuity: Let {xi }∞ i=1 , {yi }i=1 be two sequences such that x = limi→∞ xi and y = limi→∞ yi . If xi ⊥ yi for each i ∈ N, then x ⊥ y. • Homogeneity: If x ⊥ y, then λx ⊥ μy holds for any real number λ, μ ∈ R. • Symmetry: If x ⊥ y, then y ⊥ x. • Right (respectively, left) existence: For any x, y ∈ X there exists a real number α such that x ⊥ αx + y (respectively, αx + y ⊥ x). • Right (respectively, left) uniqueness: For any x, y ∈ X, x = o, there exists at most one real number α such that x ⊥ αx+y (respectively, αx+y ⊥ x). • Right additivity: If x ⊥ y and x ⊥ z, then x ⊥ y + z. • Left additivity: If y ⊥ x and z ⊥ x, then y + z ⊥ x. • Right extension: If x ⊥ y, then there exists a closed hyperplane H ⊂ X such that y ∈ H and x ⊥ H. • Left extension: If x ⊥ y, then there exists a closed hyperplane H ⊂ X such that x ∈ H and H ⊥ y. • Orthogonal diagonals: For any x, y ∈ X there exists a unique real number α such that x + αy ⊥ x − αy. It is clear that Birkhoff orthogonality and isosceles orthogonality have the nondegeneracy, simplification, and continuity property. In what follows, we discuss the other properties. 4.1. Birkhoff orthogonality 4.1.1. Characterizations. We start this section with the following important characterization of Birkhoff orthogonality. Theorem 4.1. (cf. [57, Corollary 2.2]) If x and y are any two elements of a normed linear space, then x ⊥B αx + y if and only if there exists f ∈ X ∗ (y) satisfying |f (x)| = f · x such that α = − ff (x) . Immediately we have the following corollary.

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Corollary 4.2. If x and y are two elements of a normed linear space, then x ⊥B y if and only if there exists f ∈ X ∗ \{o} such that |f (x)| = f · x and f (y) = 0. Birkhoff orthogonality can also be characterized via semi-inner products. We say that a semi-inner product is defined on a real linear space X if to any x, y ∈ X there corresponds a real number (x, y) and the following properties hold (cf. [70]): (x + y, z) = (x, z) + (y, z) and (λx, y) = λ(x, y) for all α ∈ R and all x, y, z ∈ X; (b) (x, x) > 0 for any x = o; (c) (x, y)2 ≤ (x, x)(y, y). (a)

Lumer [70] proved that a semi-inner product space is a normed linear space 1 with the norm (x, x) 2 , and the norm of any normed linear space can be generated via a semi-inner product (possibly in infinitely many ways). Giles [47] added homogeneity in the second variable to the definition of semi-inner product, namely (d)

(x, λy) = λ(x, y) for all λ ∈ R,

and he proved that the above results of Lumer remain valid. In what follows we assume that a semi-inner product satisfies properties (a)–(d). Let (·, ·) be a semi-inner product generating the norm of a normed linear space X, and let x, y ∈ X. Then x is said to be orthogonal to y in the sense of Lumer (relative to the semi-inner product (·, ·)) if (y, x) = 0; we write x ⊥L y for this case (cf. [42]). Dragomir and Koliha [42] proved the following characterization of Birkhoff orthogonality. Theorem 4.3. [42] Let X be a real normed linear space, and x, y ∈ X. Then x ⊥B y if and only if x ⊥L y relative to some semi-inner product which generates the norm of X. Observe that the above theorem says that for any semi-inner product that generates the norm, x ⊥L y implies x ⊥B y. Nevertheless, the reverse implication is generally not true (cf. Example 5.4 in [42]). Another characterization of this type is contained in the following theorem. Theorem 4.4. [42] Let X be a real normed linear space, and x, y ∈ X. Then the following facts are equivalent: (i) (ii)

x ⊥B y; For every semi-inner product (·, ·) generating the norm of X we have (y, x + μy) ≤ 0 ≤ (y, x + γy),

f or all μ < 0 < γ.

4.1.2. Homogeneity. The homogeneity of Birkhoff orthogonality follows from the absolute homogeneity of the norm.

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Theorem 4.5. Birkhoff orthogonality is homogeneous in any normed linear space. 4.1.3. Symmetry. The symmetry of Birkhoff orthogonality received much attention. Birkhoff [26] proved that if Birkhoff orthogonality is symmetric in a strictly convex normed linear space whose dimension is at least three, then the space is an inner product space. Day [37] and James [56] showed in different ways that the assumption of strict convexity in Birkhoff’s result can be released. Theorem 4.6. [37,56] A normed linear space X, whose dimension is at least three, is an inner product space if and only if Birkhoff orthogonality is symmetric in X. One of their proofs is based on the following well known result of Kakutani [64]: A real normed linear space X of dimension at least three is an inner product space if there exists a norm-one projection onto every closed linear subspace of X. The assumption on the dimension of the space in Theorem 4.6 cannot be omitted. James [56] provided examples of Minkowski planes in which Birkhoff orthogonality is symmetric (such a plane is called a Radon plane). Day [37] provided a way to construct every Radon plane, where the existence of conjugate diameters plays an essential role. Two diameters [u, v] and [s, t] of the unit circle of a Minkowski plane (i.e., chords of the unit circle passing though the origin) form a pair of conjugate diameters if u − v ⊥B s − t

and s − t ⊥B u − v.

The following theorem is of fundamental importance. Theorem 4.7. (cf. [77, Proposition 39]) The unit circle of any Minkowski plane has a pair of conjugate diameters. These diameters may be chosen so that their endpoints are extreme points of the unit disc. We refer to [77, Section 6.1.1] for historical remarks about the proofs of Theorem 4.7 and different ways to find conjugate diameters in a given Minkowski plane. Now we describe how Day [37] constructs a Radon plane from an arbitrary Minkowski plane X. By Theorem 4.7, there exists a pair of points x, y ∈ SX such that x ⊥B y and y ⊥B x, and by Corollary 4.2 there exist two linear functionals x∗ , y ∗ ∈ SX ∗ such that x∗ (x) = 1, x∗ (y) = 0, y ∗ (x) = 0, and y ∗ (y) = 1. By applying two appropriate linear isometries onto R2 , if necessary, we may assume that X is a Minkowski plane on R2 , x = x∗ = (0, 1), and y = y ∗ = (1, 0). Then the portions of SX in the first and in the third quadrant and the portions of SX ∗ in the second and in the fourth quadrant form a new curve, enclosing a closed convex region centered at the origin. And this curve

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Figure 3. Examples of unit circles of Radon planes is the unit circle of a Radon plane [37]. Figure 3 shows two examples of such constructions. For more details about the symmetry of Birkhoff orthogonality and Radon planes as well as an extension of Radon planes we refer to [74; 76; 77, Sect. 6]; see also [18]. 4.1.4. Existence. Concerning existence properties of Birkhoff orthogonality we have the following results. Theorem 4.8. (Right existence, [57]) Let X be a normed linear space. For any x, y ∈ X there exists a real number α such that x ⊥B αx + y. Moreover, such a y . If x ⊥B αx + y and x ⊥B βx + y, then x ⊥B γx + y number satisfies |α| ≤ x holds for any real number γ between α and β. The continuity of Birkhoff orthogonality and Theorem 4.8 imply that, for any x, y ∈ X, there exists a closed interval on the real line such that for each number α of this interval x ⊥B αx + y holds. James provided the following way to determine this interval. Theorem 4.9. [57] Let x = o and y be two vectors in a normed linear space, and let 1 1 lim (nx − nx + y), β = lim (nx − y − nx). α= x n→∞ x n→∞ Then α and β are the smallest and largest values of the scalar γ such that x ⊥B γx + y. ¯ y + x the relationship On the other hand, for x ⊥B αx + y and y ⊥B α between the scalars α and α ¯ is given in the next theorem. Theorem 4.10. [57, Theorem 2.4; 76, Corollary 3] Let x and y be two vectors ¯ + x then |αα ¯ | ≤ 1. in a normed linear space. If x ⊥B αx + y and y ⊥B αy

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Moreover, Birkhoff orthogonality is symmetric if and only if, for any vectors ¯ y + x, the x, y = o and scalars α, α ¯ such that x ⊥B αx + y and y ⊥B α inequality αα ¯ ≥ 0 holds. A result similar to the one in Theorem 4.8 holds for the left existence. Theorem 4.11. (Left existence, [57]) Let X be a normed linear space, x, y ∈ X. Then there exists a real number α such that αx + y ⊥B x. Moreover, αx + y = inf{βx + y : β ∈ R}. If αx + y ⊥B x and βx + y ⊥B x, then γx + y ⊥B x holds for any number between α and β. Theorem 4.1 and Corollary 4.2 show the relation between Birkhoff orthogonality and continuous linear functionals. Now we collect some other results in that framework, but dealing with closed hyperplanes through the origin denoted by H. If x ⊥B y (y ⊥B x, respectively) holds for each y ∈ H, then we say that x is Birkhoff orthogonal to H, and we denote this fact by x ⊥B H (H is Birkhoff orthogonal to y, H ⊥B y, resp.). The following theorem is an easy corollary of the Hahn–Banach theorem. Theorem 4.12. For any vector x in a normed linear space X there exists a hyperplane H ⊂ X such that x ⊥B H. On the other hand, and as the next theorem shows, if X is not finite dimensional then the existence of a point x ∈ X that is Birkhoff orthogonal to a given hyperplane H is not always guaranteed. Theorem 4.13. [58] A Banach space X is reflexive if and only if for any hyperplane H ⊂ X there exists a vector x ∈ X\{o} such that x ⊥B H. If the hyperplane is on the left-hand side, we have the following result. Theorem 4.14. [56, Theorems 4, 5] Let X be a normed linear space with dim X ≥ 3. The following facts are equivalent: (i) For each hyperplane H ⊂ X there exists x ∈ X\{o} such that H ⊥B x. (ii) For each x ∈ X there exists a hyperplane H ⊂ X such that H ⊥B x. (iii) X is an inner product space. 4.1.5. Uniqueness. In some sense, right uniqueness and left uniqueness are dual properties of each other. More precisely, we have the following results. Theorem 4.15. [57] Let X be a normed linear space. Then Birkhoff orthogonality in X is unique on the left (respectively, right) if and only if X is strictly convex (respectively, smooth). Theorem 4.16. [57, Theorem 5.3, Theorem 5.4] Let X be a normed linear space. Then the following statements hold.

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(i) If Birkhoff orthogonality is unique on the left in X ∗ , then it is unique on the right in X. If X is reflexive, the reciprocal result is also true. (ii) If Birkhoff orthogonality is unique on the right in X ∗ , then it is unique on the left in X. If SX is weakly compact, the reciprocal result is also true. Nevertheless, there are examples of normed linear spaces X where Birkhoff orthogonality is unique on the left but not unique on the right in X ∗ ; see [38]; and there are examples of normed linear spaces X where Birkhoff orthogonality is unique on the right, but not unique on the left in X ∗ ; cf. [97]. Since any Minkowski space is reflexive and its unit sphere is compact (and then weakly compact), the uniqueness of Birkhoff orthogonality on the left (respectively, right) in a Minkowski space X is equivalent to the uniqueness of Birkhoff orthogonality on the right (respectively, left) in X ∗ . 4.1.6. Additivity. Having the above theorems in mind, the next results show that additivity and uniqueness properties of Birkhoff orthogonality are closely related. Theorem 4.17. [57] Let X be a normed linear space. Then Birkhoff orthogonality in X is additive on the right if and only if X is smooth. Theorem 4.18. [56] Let X be a normed linear space. (i) If dim X = 2, then Birkhoff orthogonality is additive on the left in X if and only if X is strictly convex. (ii) If dim X ≥ 3, then Birkhoff orthogonality is additive on the left in X if and only if X is an inner product space. As with many characterizations of inner product spaces of dimension at least three, the proof of the second part of Theorem 4.18 is based on Kakutani’s theorem [64]. Recall, e.g., that this was also the case with Theorem 4.6. The additivity properties can be weakened by requiring that additivity affects only pairs of vectors which are reciprocally orthogonal. On this line, Marino and Pietramala [71] proved that a strictly convex and smooth normed linear space of dimension at least three is an inner product space if and only if Birkhoff orthogonality is left additive for biorthogonal pairs of vectors, (l.a.b.), namely, x ⊥B y, y ⊥B x, x ⊥B z, y ⊥B z ⇒ x + y ⊥B z. Later, in [4] a different proof of this characterization was given, without the requirement of strict convexity and smoothness. Theorem 4.19. [4,71] Let X be a normed linear space with dim X ≥ 3. Then X is an inner product space if and only if Birkhoff orthogonality is l.a.b. In a strictly convex Minkowski plane, Birkhoff orthogonality is additive on the left, and then evidently it is l.a.b. in such spaces. Moreover, since Birkhoff

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orthogonality is unique on the right in smooth spaces, we obtain that in a smooth Minkowski plane Birkhoff orthogonality is trivially l.a.b. In [4] an example of a non-strictly convex and non-smooth Minkowski plane is given in which Birkhoff orthogonality is l.a.b. From Theorem 4.17 it is evident that in any smooth normed linear space Birkhoff orthogonality is right additive for biorthogonal pairs of vectors, (r.a.b.), namely, x ⊥B y, y ⊥B x, z ⊥B x, z ⊥B y ⇒ z ⊥B x + y. Nevertheless, the reciprocal result is not true. There are Minkowski planes that are non-smooth and not strictly convex, and in which Birkhoff orthogonality is r.a.b.; see [4]. Moreover, since Birkhoff orthogonality is unique on the left if and only if the space is strictly convex, it follows that it is trivially r.a.b. in any strictly convex Minkowski plane. Note also that R2 endowed with the maximum norm is non-smooth and not strictly convex, and that Birkhoff orthogonality is not r.a.b. in this space. Question 4.1. Let X be a normed linear space with dim X ≥ 3 in which Birkhoff orthogonality is r.a.b. Does this imply the smoothness of X? 4.1.7. Extension. From the Hahn–Banach theorem it follows that Birkhoff orthogonality has the right extension property in any normed linear space. On the other hand, since Birkhoff orthogonality is homogeneous, it has the left extension property in any Minkowski plane. Moreover, since the left extension property implies property (ii) of Theorem 4.14, any normed linear space X, with dim X ≥ 3, is an inner product space if and only if Birkhoff orthogonality has the left extension property 4.1.8. Orthogonal diagonals and related results. Birkhoff orthogonality has the property of existence and uniqueness of orthogonal diagonals in any normed linear space, but via this property we can also characterize inner product spaces. Theorem 4.20. [19] Let X be a normed linear space, and let x, y ∈ X\{o}. Then the following statements hold. (i) There exists a unique number α := α(x, y) such that x + αy ⊥B x − αy. Moreover, such a number satisfies 3 x x ≤α≤ . 3 y y (ii) X is an inner product space if and only if, for any x, y ∈ X\{o}, the identity α(x, y) = x y holds. Since Birkhoff orthogonality is homogeneous, the second property in Theorem 4.20 says that a normed linear space X is an inner product space if

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and only if the implication x, y ∈ SX ⇒ x + y ⊥B x − y holds. Baronti [17] improved this characterization by showing that X is an inner product space if and only if the above implication holds for pairs of Birkhoff orthogonal vectors, i.e., x, y ∈ SX ,

x ⊥B y ⇒ x + y ⊥B x − y.

The fact that Birkhoff orthogonality has the property of existence of orthogonal diagonals is the key in the proof of the following debilitation of the hypothesis in the characterization of inner product spaces given by the parallelogram law. Theorem 4.21. [14,21] A normed linear space X is an inner product space if and only if the implication x, y ∈ X,

x ⊥B y ⇒ 2x2 + 2y2 ≈ x + y2 + x − y2

holds, where “≈” means either “≤” or “≥”. Up to this point we have only described the behavior of Birkhoff orthogonality in relation to the properties listed at the beginning of this section. There are many further results related to Birkhoff orthogonality, e.g., results concerning orthogonal decompositions of Banach spaces; see [1,54,99], and the references therein. We have also seen that Birkhoff orthogonality relates to characterizations of inner product spaces. More results in this direction can be found in the book of Amir [14]. We finish this section with some characterizations that appeared later. Theorem 4.22. (cf. [23,93]) Let X be a normed linear space and λ > 0 be a fixed number. Then the following statements are equivalent: (i) (ii) (iii) (iv) (v) (vi)

u, v ∈ SX , u ⊥B v ⇒ λu + v ⊥B u − λv; u − λv; u, v ∈ SX , u ⊥B v ⇒ λu + v = √ u, v ∈ SX , u ⊥B v ⇒ λu + v ≤ √1 + λ2 ; u, v ∈ SX , u ⊥B v ⇒ λu + v ≥ √1 + λ2 ; u, v ∈ SX , u ⊥B v ⇒ λu + v = 1 + λ2 ; X is an inner product space.

Theorem 4.23. [22] Let X be a Minkowski space, and let L(X) be the space of linear operators from X into itself. The following properties are equivalent: (i) (ii)

X is an inner product space; For any A, C ∈ L(X), A ⊥B C if and only if there exists u ∈ SX such that A = Au and Au ⊥B Cu.

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4.2. Isosceles orthogonality Since isosceles orthogonality is obviously symmetric, we study its properties regardless of the side. 4.2.1. Homogeneity. One of the most important properties of isosceles orthogonality is that it is homogenous only in inner product spaces. We see now the origin of this result. Theorem 4.24. [45] A normed linear space X is an inner product space if and only if, for any x, y ∈ SX and any number α, the identity αx + y = x + αy holds. Assume that isosceles orthogonality is homogeneous. Let x, y ∈ SX and α ∈ R. Since x + y ⊥I x − y, then (1 + α)(x + y) ⊥I (1 − α)(x − y), i.e., αx + y = x + αy. Thus, the above theorem yields the following one. Theorem 4.25. [55, Theorem 4.7] Isosceles orthogonality is homogeneous in a normed linear space if and only if this space is an inner product space. The hypothesis of the homogeneity of isosceles orthogonality in the above characterization can be weakened in different ways. Theorem 4.26. [69] A normed linear space X is an inner product space if and only if there exists a number α ∈ / {0, −1, 1} such that the implication x, y ∈ X,

x ⊥I y ⇒ x ⊥I αy

holds. Theorem 4.27. [2, Proposition 2.27] A normed linear space X is an inner product space if and only if there exists a number δ > 0 such that the implication x, y ∈ SX , x ⊥I y, |λ| < δ ⇒ x ⊥I λy holds. Moreover, δ can depend on x and y. 4.2.2. Additivity. From the continuity of the norm it follows easily that if isosceles orthogonality is additive then it is also homogeneous. Thus, the next theorem follows directly from Theorem 4.25. Theorem 4.28. [55, Theorem 4.8] Isosceles orthogonality is additive in a normed linear space if and only if this space is an inner product space. The uniqueness properties of isosceles orthogonality (see below) imply that Singer orthogonality (as “normalized” isosceles orthogonality) is additive in any Minkowski plane. This is not the case in higher dimensions. Theorem 4.29. [68] Let X be a normed linear space with dim X ≥ 3. Singer orthogonality is additive in X if and only if X is an inner product space.

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Since isosceles orthogonality is symmetric, the property of additivity for biorthogonal pairs of vectors, (a.b.), is stated as x ⊥I y, x ⊥I z, y ⊥I z ⇒ x ⊥I y + z, y ⊥I x + z, z ⊥I x + y. Theorem 4.30. [4] Let X be a normed linear space with dim X ≥ 3. Then X is an inner product space if and only if isosceles orthogonality is a.b. The next theorem shows that the a.b. property is trivial in two-dimensional spaces. Theorem 4.31. [4] Let X be a Minkowski plane. If x, y, z ∈ X are such that x ⊥I y, x ⊥I z and y ⊥I z, then xyz = 0. Remark 4.32. Observe that the above theorem is not true for Birkhoff orthogonality. In R2 endowed with the norm (x1 , x2 ) = max{|x1 |, |x2 |}, the vectors x = (1, −1), y = (1, 1) and z = (0, 1) satisfy x ⊥B y, x ⊥B z and y ⊥B z. 4.2.3. Existence and uniqueness. Concerning the properties of existence and uniqueness, as they were defined at the beginning of this section, isosceles orthogonality has the following behavior. Theorem 4.33. [55, Theorem 4.4; 65, Theorem 3] Let X be a normed linear space. For any x, y ∈ X there exists a number α such that x ⊥I αx + y. Isosceles orthogonality in X is unique if and only if X is strictly convex. Since isosceles orthogonality is not generally homogeneous, it makes sense to discuss this other type of existence and uniqueness property: Let X be a Minkowski plane, and let x ∈ X. For any number α > 0, check out the existence and uniqueness (up to the sign) of a point y ∈ X with y = α and x ⊥I y. The existence of the point y in the above property follows directly from the continuity of the norm. Theorem 4.34. [3] Let X be a Minkowski plane and let x ∈ X. Then for each number α > 0 there exists a point y ∈ X such that y = α and x ⊥I y. Concerning the uniqueness of y in the above theorem, we have the following results. The first one states that inside the ball with center o and radius x the vector y is unique; and the second states that the uniqueness of y outside that ball is characteristic for strictly convex spaces. Theorem 4.35. [3, Corollary 4] Let X be a Minkowski plane and let x ∈ X\{o}. For each number 0 ≤ α ≤ x there exists a unique (up to the sign) point y ∈ X with y = α and x ⊥I y. Theorem 4.36. [3, Proposition 5] Let X be a Minkowski plane. The following properties are equivalent: (i) X is strictly convex;

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For every x ∈ X\{o} and α ≥ x there exists a unique (up to the sign) point y ∈ X with y = α and x ⊥I y.

Recently, Ji, Li, and Wu [60] showed that the two types of uniqueness of isosceles orthogonality considered here are equivalent. More precisely, they showed that, if there exist two points w and z in a Minkowski plane, lying in the same open half-plane bounded by the line −x, x such that w = z , w ⊥I x, and z ⊥I x, then there exist a point y and two distinct numbers α and β such that w = αx + y and z = βx + y; for any two points x = o and y in X, if there exist two distinct numbers α1 and α2 such that x ⊥I α1 x + y and x ⊥I α2 x + y, then we have α1 x + y = α2 x + y. Following [60], assume that X is a Minkowski plane. For any point x ∈ X\{o}, denote by MX (x) the length of the longest segment contained in SX and parallel to the line −x, x, i.e., MX (x) := sup{a − b : [a, b] ⊆ SX ,

and

x (a − b) = a − b x}.

Obviously, if there is no such segment, then MX (x) = 0. The next theorem presents a sharpening of Theorem 4.35. Theorem 4.37. [60] Let X be a Minkowski plane and let x ∈ X\{o}. Then, for (α ≥ 0 when MX (x) = 0), there exists a unique each number 0 ≤ α ≤ M2x X (x) (up to the sign) point y ∈ X such that y = α and x ⊥I y. The following theorem complements Theorem 4.33. Theorem 4.38. [60] Let X be a normed linear space with dim X ≥ 2, x ∈ X\{o}, and let L ⊂ X be any two-dimensional subspace that contains x. Then, for any y ∈ L with 0 ≤ y ≤ M2x (y ≥ 0 when ML (x) = 0), there exists L (x) a unique number α such that x ⊥I αx + y. 4.2.4. Extension. The extension property implies homogeneity. Therefore, isosceles orthogonality has the extension property in a normed linear space X if and only if X is an inner product space. The next theorem, directly following from Theorem 4.29, improves this characterization when dim X ≥ 3. Theorem 4.39. Let X be a normed linear space with dim X ≥ 3. The following properties are equivalent: (i) For every x ∈ SX there exists a closed hyperplane H ⊂ X such that x ⊥I H ∩ SX . (ii) X is an inner product space. Paper [36] contains elementary proofs of related results from [55]. 4.2.5. Orthogonal diagonals and related results. It is immediate to see that for any x, y ∈ X\{o} there is a unique number α > 0 (accurately, α = x y ) such that x + αy ⊥I x − αy, i.e., any normed linear space has the property of

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existence and uniqueness of isosceles orthogonal diagonals. This fact allows is to obtain (as it was the case with Birkhoff orthogonality) another debilitation of the hypothesis in the parallelogram law. Theorem 4.40. [14,21] A normed linear space X is an inner product space if and only if the implication x, y ∈ X,

x ⊥I y ⇒ x + y2 + x − y2 ≈ 2x2 + 2y2

(∗)

holds, where “≈” means either “≤” or “≥”. It is immediate that property (∗) is equivalent to x, y ∈ SX , λ > 0,

x ⊥I λy ⇒ x + λy2 ≈ 1 + λ2 .

(∗∗)

Borwein and Keener [29] conjectured that property (∗∗) with λ fixed and “≈” equal to “=” also characterizes inner product spaces. Alonso and Ben´ıtez [7] partially solved this conjecture by showing that such a property, i.e., x, y ∈ SX ,

2

x ⊥I λy ⇒ x + λy = 1 + λ2 ,

(Pλ )

characterizes inner product spaces if λ ∈ / D, where kπ D = tan : n = 2, 3, . . . ; k = 1, 2, . . . , n − 1 . 2n On the other hand, Day [37] proved that a normed linear space X is an inner product space if and only if the Clarkson modulus of the convexity of X (see [35]), namely x + y δX (ε) := inf 1 − : x, y ∈ SX , x − y = ε , 0 ≤ ε ≤ 2, 2 satisfies the identity

δX (ε) = 1 −

1−

ε2 4

(Rε )

for every 0 ≤ ε ≤ 2. Moreover, Nordlander [85] conjectured that X is an inner product space if and only if the identity (Rε ) holds for some 0 < ε < 2. In [7] ε this conjecture was also partially answered by showing that if λ = √4−ε 2 , then the properties (Pλ ) and (Rε ) are equivalent. Moreover it was shown there that in Minkowski planes whose unit spheres are 4n-gons, n = 2, 3, . . ., the property (Pλ ) (equivalently, (Rε )) holds for λ = tan( kπ 2n ) with k = 1, 2, . . . , n − 1 [equivkπ alently, ε = 2 cos( 2n )]. Since these examples are two-dimensional, in [7] it was conjectured that if dim X ≥ 3, then (Pλ ) [equivalently, (Rε )] is characteristic for inner product spaces without restriction on λ > 0 (equivalently, 0 < ε < 2). Finally, Chelidze [32] confirmed this conjecture twenty years later by proving the following theorem.

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Theorem 4.41. [32] Let X be a normed linear space with dim X ≥ 3, and let 0 < ε < 2. Then X is an inner product space if and only if ε2 δX (ε) ≥ 1 − 1 − . (4.2) 4 Note that inequality (4.2) can be replaced by the corresponding equality since Nordlander [85] proved that for any normed linear space X we always

have that δX (ε) ≤ 1 −

1−

ε2 4 .

5. The differences between Birkhoff orthogonality and isosceles orthogonality We know that Birkhoff and isosceles orthogonalities coincide with the usual orthogonality when the space is an inner product space. Nevertheless, the above properties have shown that both concepts of orthogonality are different in a general normed linear space. In this section we collect some further results showing the difference between them even more clearly. The following characterizations of inner product spaces have been obtained by different authors and can be found in Amir’s book [14]. From them it is clear that both orthogonalities coincide only in inner product spaces. Theorem 5.1. (cf. [14, Chapter 4 and Chapter 10]) Let X be a normed linear space with unit sphere SX . Then the following properties are equivalent: (i) x, y ∈ X, x ⊥I y ⇒ x ⊥B y; (ii) x, y ∈ X, x ⊥B y ⇒ x ⊥I y; (iii) x, y ∈ SX , x ⊥I y ⇒ x ⊥B y; (iv) x, y ∈ SX , x ⊥B y ⇒ x ⊥I y; (v) x, y ∈ SX , x ⊥I y ⇒ x + y ⊥B x − y; (vi) X is an inner product space. Let X be a Minkowski plane and let u, v ∈ SX . Then it is clear that + v) ⊥I 21 (u − v) but, in general, the same does not occur for Birkhoff orthogonality. This difference can be stated in the following way. Let D(u, v) be the set of points p ∈ [u, v] such that 1 2 (u

p = inf λu + (1 − λ)v . 0≤λ≤1

Then p ⊥B 12 (u − v) for each point p ∈ D(u, v), and the difference between Birkhoff orthogonality and isosceles orthogonality can be seen from whether or not 12 (u + v) is in D(u, v). In fact, we have the following result.

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Theorem 5.2. [24] A Minkowski plane X is Euclidean if and only if there exists a real number ρ, with 1 + cos 2kπ n (2k < n; n = 3, 4, . . .), 0 < ρ < 1 and ρ = 2 such that

1

u, v ∈ SX , inf λu + (1 − λ)v = ρ ⇒ (u + v)

= ρ. 0≤λ≤1 2 (Note that Theorem 5.2 is also true with ρ = 12 ; see [25].) 5.1. Geometric consequences of the differences Once we know that Birkhoff and isosceles orthogonalities are clearly different, it is interesting to study how this difference affects geometric properties of known objects in general Minkowski spaces. In this section we show how it affects, in particular, properties of bisectors and Zindler curves in a Minkowski plane. 5.1.1. Bisectors in a Minkowski plane. The bisector B(p, q) of a line segment with endpoints p = q in a normed linear space X is defined by B(p, q) := {x ∈ X : x − p = x − q}. The concept of bisector is closely related to the construction of Voronoi diagrams and has been intensively studied in computational geometry (where most of the results are obtained without the assumption that the unit ball is centrally symmetric; cf. [15,16,66]). Bisectors of Minkowski spaces have, in general, a complicated topological and geometric structure (cf. [50–53], and the surveys [75,77]). In this subsection we show that this complicated structure can be seen as a consequence of the difference between Birkhoff orthogonality and isosceles orthogonality. Birkhoff orthogonality and isosceles orthogonality are both closely related to properties of bisectors. On the one hand, as it was shown in [14, p. 26], a p−q point z belongs to B(p, q) if and only if z − p+q 2 is isosceles orthogonal to 2 , which means that the geometric structure of bisectors in Minkowski spaces is fully determined by geometric properties of isosceles orthogonality. To see how the structure of bisectors is related to Birkhoff orthogonality, we need some notations. Let X be a Minkowski plane with a fixed orientation ω. For any x ∈ X\{o}, let Hx+ and Hx− be the two open half-planes bounded by −−−→ → = ω holds for any point z ∈ H + , and that −x, x such that (−x)z = − zx x − − − → − → = z(−x) = ω holds for any point z ∈ H − . For any x ∈ S , we denote xz X x by l(x) and r(x) the two points such that [l(x), r(x)] is a maximal segment parallel to −x, x on SX ∩ Hx+ and that r(x) − l(x) is a positive multiple of x

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Figure 4. Bisectors bounded by bent strips (Fig. 4). When there is no non-trivial segment on SX parallel to −x, x, the points l(x) and r(x) are chosen in such a way that r(x) = l(x) ∈ SX ∩ Hx+ and l(x) ⊥B x. Theorem 5.3. (cf. [75, Proposition 22]) Let X be a Minkowski plane. For any x ∈ SX , B(−x, x) is fully contained in the bent strip bounded by the rays [x, x + r(x), [x, x − l(x), [−x, −x + l(x), and [−x, −x − r(x). This theorem shows that the structure of bisectors is “somehow controlled” by properties of Birkhoff orthogonality. For the explanation of further relations between Birkhoff orthogonality and the geometric structure of bisectors we need the notion of shadow boundary. Let K be an n-dimensional convex body (i.e., a compact, convex set with nonempty interior). The shadow boundary S(K, x) of K with respect to the direction x is the union of the points of intersection of the boundary of K and each supporting line of K having direction x. Clearly, if K = BX then S(K, x) ath conconsists of all points in SX which are Birkhoff orthogonal to x. Horv´ jectured in [51] that bisectors are topological hyperplanes if and only if the corresponding shadow boundaries are (n − 2)-dimensional topological spheres. He also showed that this is true in the three-dimensional case. Theorem 5.4. [51] Let X be a three-dimensional Minkowski space. Each bisector in X is a topological 2-plane if and only if S(BX , x) is a topological 1-sphere (circle) for each point x ∈ SX . To see the difference between Birkhoff orthogonality and isosceles orthogonality in more detail we shall explore properties of the radial projection P (x) of B(−x, x), defined for any point x ∈ X\{o} by z : z ∈ B(−x, x)\{o} . P (x) := z

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It is evident that if X is the Euclidean plane, then P (x) contains precisely two points for any x ∈ X\{o} and, more generally, if X is an n-dimensional Euclidean space, then P (x) is the unit sphere of an (n − 1)-dimensional subspace. This is not true in non-Euclidean Minkowski spaces. Theorem 5.5. [78] Let X be a Minkowski plane. For any x, y ∈ SX we have that y ∈ P (x) whenever y ⊥B x. However, the next example shows that, in general, those points in SX , to which x is Birkhoff orthogonal, are not in P (x). Example 5.1. Let X be the Minkowski plane R2 with the maximum norm (α, β) = max{|α|, |β|} and x = (1, 1). Then B(−x, x) = (−1, 1), (1, −1), and therefore P (x) = {(1, −1), (−1, 1)}. It is clear that (0, 1) ∈ P (x) and (1, 0) ∈ P (x), while x ⊥B (0, 1) and x ⊥B (1, 0). Theorem 5.5 can be used to show that in a non-Euclidean Minkowski plane there exists at least one bisector which is not a straight line (of course, this is also a consequence of the fact that isosceles orthogonality is not homogeneous in such a space). Namely, let X be a non-Euclidean Minkowski plane. From Theorem 5.1 it follows that there exist two points x, y ∈ SX such that y ⊥I x but y ⊥B x. There also exists a point y ∈ SX such that y ⊥B x, and therefore y = ±y. Since y ⊥I x, we have that ±y ∈ P (x), and from Theorem 5.5 it follows that ±y ∈ P (x), which implies that B(−x, x) cannot be a straight line. Theorem 5.6. [78] Let X be a Minkowski plane and x ∈ SX . If there exists a unique point z ∈ SX (up to the sign) such that x ⊥B z, then z ∈ P (x). And if there exists a point z ∈ P (x)\P (x), then either z ⊥B x or x ⊥B z. In [78] it is shown that, in general, the condition y ∈ SX together with y ⊥B x does not imply that y ∈ P (x); the condition that z ∈ SX is the unique point (up to the sign) satisfying x ⊥B z do not imply z ∈ P (x). In addition, the arc P (x) ∩ Hx+ of SX (possibly degenerate to a point) has nothing to do with the points that are Birkhoff orthogonal to x or with the points to which x is Birkhoff orthogonal (see Remark 2.10 in [78] and the examples provided there). Then it is clear that P (x) is in general not a closed set. And we can construct strictly convex and smooth Minkowski planes such that there exists a point x ∈ SX with the property that P (x) is not closed. Thus we can check the following conjecture. Conjecture 5.1. For any non-Euclidean Minkowski plane X, there exists a point x ∈ SX such that P (x) is not closed. It follows from Theorem 5.6 that if P (x) is not closed for a point x ∈ SX then there exists a point y ∈ SX with x ⊥B y or y ⊥B x such that y ∈ P (x).

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However, x ⊥B y and y ∈ P (x) do not imply that P (x) is not closed. Thus one can also verify the following related conjecture. Conjecture 5.2. For any non-Euclidean Minkowski plane X, there exist two points x, y ∈ SX satisfying x ⊥B y or y ⊥B x such that y ∈ P (x). The following theorem is also proved in [78]. Theorem 5.7. [78] A Minkowski plane X is Euclidean if and only if for any x, y ∈ SX with x = ±y, P (x) ∩ P (y) = ∅. In other words, in any non-Euclidean Minkowski plane X there exist points x, y ∈ SX , with x = ±y, such that P (x) ∩ P (y) = ∅. Clearly, P (x) ∩ P (y) = ∅ does not imply B(−x, x) ∩ B(−y, y)\{o} = ∅ and we come to the following conjecture. Conjecture 5.3. In any non-Euclidean Minkowski plane X, there exist two points x, y ∈ SX , with x = ±y, such that B(−x, x) ∩ B(−y, y)\{o} = ∅. Finally we note that, due to the complicated structure of bisectors in Minkowski planes it also makes sense to provide sufficient conditions for a planar curve or a point set to be a bisector for a suitable Minkowski plane. 5.1.2. Zindler curves in Minkowski planes. By a curve in a normed linear space X we mean the range of a continuous function φ that maps a closed bounded interval [α, β] into X. Furthermore, a curve defined by φ : [α, β] → X is called closed if [α, β] is replaced by a Euclidean circle, say, and it is simple if it has no self-intersections. Moreover, such a curve C is said to be rectifiable if the set of all Riemann sums n φ(γi ) − φ(γi−1 ) : (γ0 , γ1 , . . . , γn ) is a partition of [α, β] i=1

with respect to the norm · of X is bounded from above. If C is rectifiable, then we denote by |C| its length, i.e., n |C| := sup φ(γi ) − φ(γi−1 ) : (γ0 , γ1 , . . . , γn ) is a partition of [α, β] . i=1

A parametrization c of C is said to be regular if it has non-vanishing one-sided derivatives everywhere, and the curve C is said to be regular if it admits a regular parametrization. We denote by C the set of simple, regular, rectifiable, closed curves, which are piecewise continuously differentiable and have one-sided derivatives. In this section each curve C is planar, and it is either from the family C or a closed convex curve (i.e., the boundary of a compact, convex set with nonempty interior). Two points p, q ∈ C are said to form a (Minkowskian) halving pair of C if they split C regarding its Minkowskian length into two equal parts,

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and the distance p − q between them is called the corresponding (Minkowskian) halving distance. A closed curve is said to be a (Minkowskian) Zindler curve if it is of constant halving distance. A parametrization c : [0, |C|) → C of C is said to be a halving pair parametrization if every pair of points c(γ) and c(γ + 12 |C|)) is a halving pair of the curve C. For a Minkowski plane X, there exists a unique (up to a scalar factor) Haar measure on X (cf. [96, Sect. 1.4]). Thus we may assume that the underlying Minkowski plane is endowed with a Euclidean structure, and therefore we can use the corresponding Lebesgue measure to calculate Minkowskian areas. A chord of a closed convex curve C bisecting the area of conv(C) is called an area-halving chord, and the (Minkowskian) length of such a chord is said to be the corresponding area-halving distance. Let C be a closed convex curve. A parametrization c : [0, |C|) → C of C is an area-bisecting parametrization if every chord [c(γ), c(γ + 12 |C|)] is an area-halving chord. In the Euclidean case, Zindler curves are not necessarily circles, have many interesting characterizations and are strongly related to other concepts, such as curves of constant area-halving distance and curves of constant width (see Section 2 of the survey [75]). Zindler curves are also related to the construction of graphs of low geometric dilation; see [43,44,48] for investigations of the geometric dilation problem in the Euclidean plane, and [79] for the extension of this problem to Minkowski planes. Zindler [101, Sect. 7] (see also [43, Theorem 4]) proved that the following statements are equivalent for a closed, convex curve C in the Euclidean plane: 1. All halving chords of C have the same length. 2. All chords of C bisecting the area have the same length. 3. The halving chords and the area-halving chords of C coincide. In the following we show that, due to the difference between Birkhoff orthogonality and isosceles orthogonality, this result does not hold in general Minkowski planes. Let c : [0, |C|) → C be a parametrization of a closed curve C. The curve Mc (which is called the midpoint curve of C) and the curve Cc∗ corresponding to C with respect to c are defined by the parametrizations

1 1 1 1 and c∗c (γ) := , mc (γ) := c(γ)+c γ + |C| c(γ)−c γ + |C| 2 2 2 2 respectively. It is clear that the curve Cc∗ is centrally symmetric and that if c is piecewise continuously differentiable, then both parametrizations are piecewise continuously differentiable. We also use the arc-length parametrization c¯ : [0, |C|) → C of a rectifiable closed curve C, which is continuous, bijective, and has the property that c(γ) ˙ = 1, whenever the derivative exists. As in the Euclidean case, one can prove that every curve C ∈ C in an arbitrary Minkowski plane admits an arc-length parametrization which is piecewise continuously differentiable.

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The following lemma shows the relation between isosceles orthogonality and halving pair parametrizations. Lemma 5.8. [83] Let c : [0, |C|) → C be a piecewise continuously differentiable parametrization of a closed curve C ∈ C. The parametrization c is a halving pair parametrization of C if and only if c˙∗c (γ) ⊥I m˙ c (γ), whenever the derivatives exist. Next we provide a characterization of Zindler curves based on Birkhoff orthogonality. Theorem 5.9. [83] Let c¯ : [0, |C|) → C be an arc-length parametrization of C which is piecewise continuously differentiable. Then C is a Zindler curve if and only if c∗c¯ (γ) ⊥B c˙∗c¯ (γ), whenever the derivative exists. Now we turn to the characterization of closed convex curves of constant area-halving distances. Clearly, a chord [c(γ), c(γ + 12 |C|)] bisects the Euclidean area of conv(C) if and only if it bisects the Minkowskian area of conv(C). Thus Lemma 3.5 from [48], referring to the Euclidean case, can be carried over to Minkowski planes, and we have the following characterization of areabisecting parametrization. Lemma 5.10. (based on [48, Lemma 3.5]) Let c : [0, |C|) → C be a piecewise continuously differentiable parametrization of a closed convex curve C in a Minkowski plane. It is an area-bisecting parametrization if and only if m ˙ c (γ) ˙ and c(γ ˙ + 12 |C|) exist. is parallel to c∗c (γ), whenever the derivatives c(γ) Here comes the first characterization of closed convex curves of constant area-halving distance. Lemma 5.11. [83] Let c : [0, |C|) → C be a piecewise continuously differentiable area-bisecting parametrization of a closed convex curve C. Then C is a curve of constant area-halving distance if and only if c∗c (γ) ⊥B c˙∗c (γ), whenever the derivative exists. Lemmas 5.10 and 5.11 imply the following theorem. Theorem 5.12. [83] Let c : [0, |C|) → C be a piecewise continuously differentiable area-bisecting parametrization of a closed convex curve C. Then C is a curve of constant area-halving distance if and only if we have m ˙ c (γ) ⊥B c˙∗c (γ), whenever the derivatives c(γ) ˙ and c(γ ˙ + 12 |C|) exist. As we have shown before, Birkhoff orthogonality and isosceles orthogonality are different orthogonality types in non-Euclidean Minkowski planes. Moreover, there exist a Minkowski plane X and two points x, y ∈ SX with x ⊥B y such that y is the unique point (up to the sign) in SX to which x is Birkhoff orthogonal, and that x ⊥I ty holds for any number t = 0 (see Remark 2.10 in [78]). Thus it follows from Lemmas 5.8 and 5.10 that a halving pair

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parametrization of a closed convex curve C of constant halving distance is not necessarily an area-bisecting parametrization of C. Similarly, an area-bisecting parametrization of a closed convex curve of constant area-halving distance is not necessarily a halving pair parametrization of C. The following natural problems are posed in [83]: 1. Find a general way for the construction of Zindler curves in Minkowski planes. 2. Characterize midpoint curves of Zindler curves. 3. Find sets A, B ⊂ X such that the following implication characterizes inner product spaces: x ∈ A, y ∈ B, x ⊥I y ⇒ x ⊥B y. We note that the first problem could be very difficult since there is still no general way to construct Zindler curves in the Euclidean plane. The solution to the second problem might be very important for solving the first one since many of the existing constructions of Zindler curves in the Euclidean plane start from a midpoint curve. The third problem is useful to study the relation between convex Zindler curves and closed convex curves of constant area-halving distance. 5.2. Quantitative characterizations of the differences Having established in the previous sections that Birkhoff and isosceles orthogonalities are different, it seems natural to try to measure such a difference for a given normed linear space. For example, it is intuitively clear that the 2 than in the Euclidean plane l22 (in which case, obvidifference is “larger” in l∞ ously, the difference is vanishing). In [61] such a measurement was provided by introducing the geometric constant D(X) = inf inf x + λy : x, y ∈ SX , x ⊥I y . λ∈R

Since inf λ∈R x + λy ≤ 1 for any points x, y ∈ SX and the equality holds if and only if x ⊥B y, it is clear that D(X) measures the worst-case difference between Birkhoff orthogonality and isosceles orthogonality. The following theorem presents general bounds on this constant. Theorem 5.13. [61] For any normed linear space X, we have √ 2( 2 − 1) ≤ D(X) ≤ 1. Moreover, D(X) = 1 if and only if X is an inner product space. The lower bound of D(X) is attained in any normed linear space that contains a two-dimensional subspace whose unit circle is a parallelogram. But

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the next theorem shows that these spaces are not the only ones for which √ D(X) = 2( 2 − 1). Theorem 5.14. [61] Let X be a normed linear space. The following properties are equivalent: √ (i) There exist x, y ∈ SX such that x ⊥I y and inf λ∈R x + λy = 2( 2 − 1). (ii) There exist a two-dimensional subspace X0 of X and a point x0 ∈ SX0 such that x0 is the common endpoint of two segments contained in SX0 √ whose lengths are not less than 2. A Minkowski plane X is said to be symmetric if there exist e1 , e2 ∈ SX such that e1 + te2 = e1 − te2 = e2 + te1 = e2 − te1 holds for any t ∈ R. The pair {e1 , e2 } is a pair of axes of X. The Minkowski planes whose unit circle is a parallelogram are particular cases of symmetric Minkowski planes. Also, all lp2 planes are symmetric Minkowski planes. The next theorem shows how to compute D(X) for such spaces. Theorem 5.15. [61] Let X = (R2 , ·) be a symmetric Minkowski plane and let e1 = (1, 0), e2 = (0, 1) be a pair of axes of X. Then 1 + t2 D(X) = inf , t∈R (t, 1) (t, 1)∗ where ·∗ denotes the norm of X ∗ . is a parallelogram, then the value of D(X) It is easy to confirm that if SX √ in Theorem 5.15 coincides with 2( 2 − 1). From Theorem 5.15, the following corollary immediately follows. Corollary 5.16. [61] If X is a symmetric Minkowski plane, then D(X) = D(X ∗ ). Generally, it is not easy to get the exact value of D(X) for an arbitrary normed linear space X. Nevertheless, it is not an obstacle to be able to affirm that, in general, D(X) = D(X ∗ ). Example 5.2. Let X be the Minkowski plane R2 endowed with the norm whose unit sphere SX is the hexagon with √ √ p1 = (1, 1), p2 = (1 − 2, 1), p3 = (−1, 2 − 1), and their opposites as vertices (see Fig.√5). Since [−p3 , p1 ] and [p1 , p2 ] are two segments of√SX with length equal to 2, it follows from Theorem 5.14 that D(X) = 2( 2 − 1). On the other hand, the unit sphere SX ∗ of X ∗ is the hexagon with √ √ q1 = (1, 0), q2 = (0, 1), q3 = (−1/ 2, 1/ 2),

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√ Figure 5. D(X) = 2( 2 − 1) < D(X ∗ ) and their opposites as vertices. Since SX ∗ does not √ contain two consecutive 2, it follows that D(X ∗ ) > segments both of length greater than or equal to √ 2( 2 − 1) = D(X). In [59], another geometric constant to measure the difference between these two types of orthogonality was introduced, by defining D (X) := sup{x + y − x − y : x, y ∈ SX , x ⊥B y}. Theorem 5.17. [59] For any real normed linear space X we have 0 ≤ D (X) ≤ 1. Moreover, D (X) = 0 if and only if X is an inner product space. In [59] a characterization of the case when D (X) = 1 is attained at points of SX was also given but, unfortunately, it was not correct. Here is the correct statement. Theorem 5.18. Let X be a normed linear space, and let x, y ∈ SX . The following properties are equivalent: (i) x ⊥B y and x + y − x − y = 1. (ii) The segments [x, y] and [x, x − y] are contained in SX . Proof. (i)⇒(ii) Let x, y ∈ SX satisfy x ⊥B y and x + y − x − y = 1. Since x + y ≤ 2 and x − y ≥ 1, we have that x + y = 2 and x − y = 1. Therefore x, y, 12 (x + y) ∈ SX , which implies that [x, y] ⊂ SX . Moreover, for each λ ∈ [0, 1], 1 = x ≤ x + (λ − 1)y = λx + (1 − λ)(x − y) ≤ λ x + (1 − λ) x − y = 1, which implies that λx + (1 − λ)(x − y) = 1, and then [x, x − y] ⊂ SX . (ii)⇒(i) Let x, y ∈ SX such that [x, y], [x, x − y] ⊂ SX . Then it is clear that x + y = 2 and x − y = 1, and we only need to show that x ⊥B y. If

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Figure 6. D (X) = 1 > D (X ∗ ) λ ∈ [−1, 0], then x + λy = (−λ)(x − y) + (1 − (−λ))x = 1, and if λ ∈ [0, 1], we have

1 λ

= 1 + λ ≥ 1. x+ y

x + λy = (1 + λ)

1+λ 1+λ

Thus x + λy ≥ 1 = x for any λ ∈ [−1, 1] (and therefore for any λ ∈ R, since the function λ → x + λy is convex), which implies that x ⊥B y. With the help of Theorem 5.18, our next example shows that, in general, D (X) = D (X ∗ ). Example 5.3. Let X be the Minkowski plane R2 endowed with the norm whose unit sphere SX is the octagon with p1 = (1, 0),

p2 = (0, 1),

p3 = (−1/2, 6/5),

p4 = (−1, 1),

and their opposites as vertices (see Fig. 6). Since [p1 , p2 ] and [p1 , p1 − p2 ] = [p1 , −p4 ] are two segments of SX , it follows from Theorem 5.18 that D (X) = 1. On the other hand, the unit sphere SX ∗ of X ∗ is the octagon with q1 = (1, 0),

q2 = (1, 1),

q3 = (2/5, 1),

q4 = (−2/7, 5/7),

and their opposites as vertices. Since there are no x, y ∈ SX ∗ satisfying property (ii) of Theorem 5.18, it follows that D (X ∗ ) < 1 = D (X). At the end of this section we present a result on the relation between D(X) and D (X) when X is a Minkowski space. It is based on the attainment of the extreme values of their respective ranges. Theorem 5.19. Let X be a Minkowski space. Then the following statements hold. = 0. (i) D(X) = 1 if and √ only if D (X) (ii) If D(X) = 2( 2 − 1) then D (X) = 1.

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Proof. (i) This is obvious since in both cases X must be an inner product space. √ (ii) Assume that X is finite dimensional and D(X) = 2( 2 − 1). Then it follows from Theorem 5.14 that there exist a two-dimensional subspace X0 of X and three points √ w, x, z ∈ SX0 such that [w, x], [x, z] ⊂ SX0 and x − w = x − z = 2. Let e1 ∈ [w, x] and e2 ∈ [x, z] be such that x − e1 = x − e2 = 1. Then e1 and x − e2 are points of SX and both are in the same semi-plane defined by the line −x, x. Since the distance from these points to x is equal to 1 and the length of [w, x] is larger than 1, it follows from the Monotonicity Lemma [77, Proposition 31] that e1 = x − e2 . Taking y = e1 , it follows from Theorem 5.18 that x ⊥B e1 and x + e1 − x − e1 = 1, and therefore, D (X) = 1. Note that the reciprocal of property (ii) in Theorem 5.19 is not true. √ The space X in Example 5.3 satisfies D (X) = 1, whereas D(X) > 2( √ 2 − 1) because SX contains no segment of length greater than or equal to 2.

6. Connections between Birkhoff orthogonality and isosceles orthogonality In this section we recall some results that show the existing connection between Birkhoff orthogonality and isosceles orthogonality. In this direction, R.C. James proved the following theorem. Theorem 6.1. [57, Theorem 3.4] If x and y are any two points of a normed linear space X, and the numbers αn (n = 1, 2, . . .) are such that x ⊥I

1 (αn x + y), n

then α := limn→∞ αn exists and x ⊥B αx + y. Furthermore, α is the mean of the largest and smallest of the numbers β for which x ⊥B βx + y. Another way to find connections between these orthogonalities is to study properties of maps preserving orthogonality types. A map T is said to be orthogonality preserving if the property x ⊥ y implies T (x) ⊥ T (y). It is known (see, e.g. [33]) that an orthogonality preserving linear map between two inner product spaces is necessarily a scalar multiple of a linear isometry. Blanco and Turˇ nsek [27] extended this result to (real or complex) normed linear spaces for the case of Birkhoff orthogonality. Theorem 6.2. [27] Let X and Y be two normed linear spaces. A linear map T : X → Y preserves Birkhoff orthogonality if and only if it is a scalar multiple of a linear isometry.

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A special case of Theorem 6.2, namely when X = Y and X is real, was obtained in [67]. In [81] it was shown that this type of results also shows connections between Birkhoff orthogonality and isosceles orthogonality. Theorem 6.3. [81] Let X and Y be two real normed linear spaces. If a linear map T : X → Y preserves isosceles orthogonality, then it also preserves Birkhoff orthogonality. Theorems 6.3 and 6.2 imply Theorem 6.4. [34,81] Let X and Y be two real normed linear spaces. A linear map T : X → Y preserves isosceles orthogonality if and only if T is a scalar multiple of a linear isometry. Chmieli´ nski and W´ ojcik [34] proved Theorem 6.4 independently by studying the stability of the isosceles-orthogonality preserving property. Mojˇskerc and Turnˇsek [84] studied properties of maps approximately preserving Birkhoff orthogonality. Theorem 6.4 can also be proved by using another approach which was used to obtain Theorem 3 of [94] (where the two-dimensional case was not considered). This approach can be summarized as follows: first one shows that a linear map f : X → Y preserving isosceles orthogonality also preserves isosceles triangles, which in turn will imply that this map preserves “equality of distance”, i.e., there exists g : [0, +∞) → [0, +∞) such that for any x, y ∈ X, f (x) − f (y) = g(x − y). Then, by the main result of [98], the map has to be a scalar multiple of a linear isometry. Although Birkhoff orthogonality and isosceles orthogonality are different even on the unit sphere of a normed linear space, we can always find a pair of points x, y in the unit circle of a Minkowski plane such that x ⊥I y and x ⊥B y. To prove this fact, we need the following lemma. Lemma 6.5. Let X be a Minkowski plane and let u, v, u , v ∈ SX be such that u + v = u + v . Assume that u = ±v and u = αu + βv with α ≥ 0, β ≥ 0. Then we have the following possibilities, according to the location of u : (1) If 0 < α ≤ β ≤ 1, then α + β = 1 and [u, v ] ⊂ [u, u ] ⊂ [u, v] ⊂ SX . (2) If 0 < β ≤ α ≤ 1, then α + β = 1 and [u, u ] ⊂ [u, v ] ⊂ [u, v] ⊂ SX . (3) If 0 < α ≤ 1 ≤ β, then β − α = 1 and [u , v] ⊂ [u , −v ] ⊂ [u , −u] ⊂ SX . (4) If 0 < β ≤ 1 ≤ α, then α − β = 1 and [u , u] ⊂ [u , −v ] ⊂ [u , −v] ⊂ SX . (5) If 1 ≤ α ≤ β, then β − α = 1 and [u , −v ] ⊂ [u , v] ⊂ [u , −u] ⊂ SX . (6) If 1 ≤ β ≤ α, then α − β = 1 and [u , −v ] ⊂ [u , u] ⊂ [u , −v] ⊂ SX . Proof. We give the proof only for the cases (2) and (4) that will be used in Theorem 6.6. The other cases are analogous. (2) Since u ∈ SX , it follows that 1 ≤ α +β. Then 1 = v = u +v −u = (1 − α)u + (1 − β)v ≤ 1 − α + 1 − β ≤ 1, from which it follows that α + β = 1. Thus v is an interior point of [u, v], which implies that [u, v ] ⊂ [u, v] ⊂ SX .

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Since 0 < β ≤ α ≤ 1 = α + β, it follows that β < 1, and from the identity 1−α u = ( α−β 1−β )u + ( 1−β )v we get that u ∈ [u, v ]. (4) Since 1 = v = (1 − α)u + (1 − β)v ≤ α − 1 + 1 − β = α − β = (α − β)u = (1 − β)u − βv ≤ 1 − β + β = 1, it follows that α − β = 1. Then u = (1 − β)u + β(−v ), which implies that u is an interior point of [u , −v ], 1 and then [u , u] ⊂ [u , −v ] ⊂ SX ; and −v = ( α−1 α )u + α (−v), which implies [u , −v ] ⊂ [u , −v] ⊂ SX . Theorem 6.6. Let X be a Minkowski plane. Then there exist two points x, y ∈ SX such that x ⊥I y and x ⊥B y. Proof. Let ω be a fixed orientation in X. It was proved in [62] and [5], respectively, that the sets → = ω} uv S := {u + v : u, v ∈ S , u ⊥ v, − B

X

B

and SI := {u + v : u, v ∈ SX , u ⊥I v} are Jordan rectifiable curves that enclose two times the area enclosed by SX . Since these curves are centered at the origin, they have to intersect in a point →= xy z ∈ X. Then there exist x, y, x , y ∈ SX such that x + y = x + y = z, − −− → x y = ω, x ⊥B y and x ⊥I y . Let us show that x = x and y = y . Let α, β ∈ R be such that x = αx+βy. Then y = (1−α)x+(1−β)y. Since x ⊥B y, we have that 1 = αx + βy ≥ |α| and 1 = (1 − α)x + (1 − β)y ≥ −− → →=x y = ω, we have that β ≤ α. |α − 1|. Therefore, 0 ≤ α ≤ 1. Since − xy Now we will see that β ≥ 0. Assume, on the contrary, that β < 0. Taking 1−β −β and β = α−β we have 0 < β ≤ 1 ≤ α . Moreover, x = α x + β y . α = α−β Taking u = x , v = y , u = x and v = y, it follows from Lemma 6.5(4) that α − β = 1 and [x, x ] ⊂ [x, −y] ⊂ [x, −y ] ⊂ SX . Then α − β = 1 and 2 = x − y = x + y = x + y. Moreover, 2(1 − β) = (1 − β)(x + y) = y + (α − β)x ≤ 1 + α − β = 2, from which β ≥ 0 follows, against the assumption. Therefore 0 ≤ β ≤ α ≤ 1. If β = 0, then x = x and y = y follows. Otherwise, if β > 0, then taking u = x, v = y, u = x , v = y , it follows from Lemma 6.5(2) that [x, x ] ⊂ [x, y ] ⊂ [x, y] ⊂ SX . Then 2 = x +y = x −y and 2 ≥ x − y = x − x + x − y + y − y, which implies x = x and y = y . This contradicts the hypothesis that β > 0.

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Javier Alonso Departamento de Matem´ aticas Universidad de Extremadura 06006 Badajoz Spain e-mail: [email protected] Horst Martini Fakult¨ at f¨ ur Mathematik Technische Universit¨ at Chemnitz 09107 Chemnitz Germany e-mail: [email protected]

Vol. 83 (2012)


Senlin Wu Department of Applied Mathematics Harbin University of Science and Technology 150080 Harbin China e-mail: senlin [email protected] Received: June 9, 2011

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