The geometry of Minkowski spaces — A survey. Part I

Martini, Horst; Swanepoel, Konrad J.; Weiss, Gunter

doi:10.1016/s0723-0869(01)80025-6

Cited by 195 publications

(189 citation statements)

References 126 publications

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“…A [11,Section 5] and [17,Section 2]. Equivalently, the latter condition can be given as the equality DK = λ · B.…”

Section: Theorem 1 Let Bmentioning

confidence: 99%

On planar convex bodies of given Minkowskian thickness and least possible area

Averkov

2005

Arch. Math.

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Assume that the width of a planar convex body K at every direction u is not less than the width of a fixed centrally symmetric body B at the direction u. We present a precise description of those bodies K given above that have least possible area. Introduction.By E d , d 2, we denote the d-dimensional Euclidean space with origin o and norm | . |. Length and volume in E d are denoted by µ and V , respectively. The unit sphere in E d is denoted by S d−1 . A set K E d is said to be a convex body if it is convex, compact and has non-empty interior, cf.[4] and [21]. If u is ranging over E d , then the functions h K (u) := max{ x, u : x ∈ K} and w K (u) := h K (u)+h K (−u) are called the support function and the width function of K, respectively (cf.[16] and [9]). For u ∈ S d−1 , w K (u) is the width of K at direction u, i.e., the distance between the two different supporting hyperplanes of K being orthogonal to u. Let us fix an arbitrary convex body B E d centered at the origin. The object of our investigation is the class X (B) of convex bodies K E d satisfying the constraint w K (u) h B (u) for any u ∈ S d−1 and minimizing the volume. We introduce the Minkowskian thickness B (K) of a convex body K E d with respect to the gauge B by the equality B (K) := min{w K (u)/ h B (u) : u ∈ S d−1 }. If B is the Euclidean unit ball, then B (K) is the usual thickness (= minimal width). Obviously, we can describe X (B) as the class of convex bodies having Minkowskian thickness one and least possible volume. Furthermore, it is easy to see that the homothetical copies of bodies from X (B) correspond to the equality case in the geometric inequalityan arbitrary convex body and α(B) is the volume of convex bodies from X (B). The sharp estimates for the quantity α(B) are found in [1]. We cite the books [4, §10], [12, §6], [5], [6], and [22, Sections 4.4 and 4.5], where various geometric inequalities in Minkowski and Euclidean spaces are discussed. If d = 2 and B is the Euclidean unit disk, then the elements of X (B) are the planar convex bodies of Euclidean thickness one and least possible area. It was proved by Pál that for the above case X (B) is the class of equilateral

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“…A [11,Section 5] and [17,Section 2]. Equivalently, the latter condition can be given as the equality DK = λ · B.…”

Section: Theorem 1 Let Bmentioning

confidence: 99%

On planar convex bodies of given Minkowskian thickness and least possible area

Averkov

2005

Arch. Math.

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“…Thus, we write (R 2 , U ) for a Minkowski plane with unit ball U , whose boundary is the unit circle of (R 2 , U ). The geometry of normed planes and spaces, usually called Minkowski Geometry (see [21], [14], and [13]), is strongly related to and influenced by the fields of Convexity, Banach Space Theory, Finsler Geometry and, more recently, Discrete and Computational Geometry. The present paper can be considered as one of the possibly first contributions to Discrete Differential Geometry in the spirit of Minkowski Geometry.…”

Section: Introductionmentioning

confidence: 99%

Involutes of polygons of constant width in Minkowski planes

Craizer

Martini²

2015

Ars Math. Contemp.

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Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and −P . Then the polygon U , as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of constant U -width, and we prove that many properties of the smooth case, which is already completely studied, are preserved. The iteration of involutes generates a pair of sequences of polygons of constant width with respect to the Minkowski norm and its dual norm, respectively. We prove that these sequences are converging to symmetric polygons with the same center, which can be regarded as a central point of the polygon P .

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“…83 (2012) On Birkhoff orthogonality and isosceles orthogonality 155 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]).…”

Section: Introductionmentioning

confidence: 99%

On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces

2011

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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.

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The geometry of Minkowski spaces — A survey. Part I

Cited by 195 publications

References 126 publications

On planar convex bodies of given Minkowskian thickness and least possible area

On planar convex bodies of given Minkowskian thickness and least possible area

Involutes of polygons of constant width in Minkowski planes

On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces

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