Reduced convex bodies in Euclidean space—A survey

Lassak, Marek; Martini, Horst

doi:10.1016/j.exmath.2011.01.006

Cited by 28 publications

(20 citation statements)

References 35 publications

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“…In analogy to the definition of reduced bodies in Euclidean space E d introduced in [7] (see also [8][9][10] and [11]), we define reduced convex bodies on S d . We…”

Section: Proposition 1 Let Diam(c) >mentioning

confidence: 99%

“…This and the assumption that K is an arbitrary hemisphere There are more questions on spherical reduced bodies. For instance, which properties of reduced bodies in E d , and especially in E 2 (see [9] and [10]), may be reformulated and proved for reduced bodies on [11]). By Theorem 4 we obtain the following spherical analog of a theorem from [8], see also Corollary 1 in [9] and [10].…”

Section: Diametermentioning

confidence: 99%

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Width of spherical convex bodies

Lassak

2013

Aequat. Math.

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170

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Abstract. For every hemisphere K supporting a convex body C on the sphere S d we define the width of C determined by K. We show that it is a continuous function of the position of K. We prove that the diameter of every convex body C ⊂ S d equals the maximum of the widths of C provided the diameter of C is at most π 2. In a natural way, we define spherical bodies of constant width. We also consider the thickness Δ(C) of C, i.e., the minimum width of C. A convex body R ⊂ S d is said to be reduced if Δ(Z) < Δ(R) for every convex body Z properly contained in R. For instance, bodies of constant width on S d and regular spherical odd-gons of thickness at most π 2 on S 2 are reduced. We prove that every reduced smooth spherical convex body is of constant width. Mathematics Subject Classification (2010). 52A55, 97G60.

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“…In analogy to the definition of reduced bodies in Euclidean space E d introduced in [7] (see also [8][9][10] and [11]), we define reduced convex bodies on S d . We…”

Section: Proposition 1 Let Diam(c) >mentioning

confidence: 99%

Section: Diametermentioning

confidence: 99%

Width of spherical convex bodies

Lassak

2013

Aequat. Math.

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170

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“…in Pál's problem ;Heil 1978). More information on reduced convex bodies in Euclidean space can be found in the survey (Lassak and Martini 2011).…”

Section: Background and Motivationmentioning

confidence: 99%

Vertex-facet assignments for polytopes

Jähn

Winter

2020

Beitr Algebra Geom

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Motivated by the search for reduced polytopes, we consider the following question: For which polytopes exists a vertex-facet assignment, that is, a matching between vertices and non-incident facets, so that the matching covers either all vertices, or all facets? We provide general conditions for the existence of such an assignment. We conclude that such exist for all simple and simplicial polytopes, as well as all polytopes of dimension $$d\le 6$$ d ≤ 6 . We construct counterexample in all dimensions $$d\ge 7$$ d ≥ 7 .

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“…Each of the 2 d+1 parts of S d dissected by d + 1 pairwise orthogonal (d − 1)dimensional spheres of S d is a spherical body of constant width π 2 , which easily follows from the definition of a body of constant width. The class of spherical bodies of constant width is a subclass of the class of spherical reduced bodies considered in [6] and [8], and mentioned by [3] in a larger context, (recall that a convex body R ⊂ S d is called reduced if Δ(Z) < Δ(R) for every body Z ⊂ R different from R, see also [7] for this notion in E d ).…”

Section: Spherical Bodies Of Constant Widthmentioning

confidence: 99%

Spherical bodies of constant width

Lassak

Musielak

2018

Aequat. Math.

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The intersection L of two different non-opposite hemispheres G and H of the ddimensional unit sphere S d is called a lune. By the thickness of L we mean the distance of the centers of the (d − 1)-dimensional hemispheres bounding L. For a hemisphere G supporting a convex body C ⊂ S d we define width G (C) as the thickness of the narrowest lune or lunes of the form G ∩ H containing C. If width G (C) = w for every hemisphere G supporting C, we say that C is a body of constant width w. We present properties of these bodies. In particular, we prove that the diameter of any spherical body C of constant width w on S d is w, and that if w < π 2 , then C is strictly convex. Moreover, we check when spherical bodies of constant width and constant diameter coincide. Mathematics Subject Classification. 52A55.A spherical ball B ρ (x) of radius ρ ∈ (0, π 2 ], or a ball for short, is the set of points of S d at distances at most ρ from a fixed point x, which is called the center of this ball. An open ball (a sphere) is the set of points of S d having distance smaller than (respectively, exactly) ρ from a fixed point. A spherical ball of radius π 2 is called a hemisphere. So it is the common part of S d and a closed half-space of E d+1 . We denote by H(m) the hemisphere with center m. Two hemispheres with centers at a pair of antipodes are called opposite.A spherical (d − 1)-dimensional ball of radius ρ ∈ (0, π 2 ] is the set of points of a (d − 1)-dimensional great sphere of S d which are at distances at most ρ from a fixed point. We call it the center of this ball. The (d − 1)-dimensional balls of radius π 2 are called (d − 1)-dimensional hemispheres, and semicircles for d = 2.A set C ⊂ S d is said to be convex if no pair of antipodes belongs to C and if for every a, b ∈ C we have ab ⊂ C. A closed convex set on S d with nonempty interior is called a convex body. Some basic references on convex bodies and their properties are [4], [9] and [10]. A short survey of other definitions of convexity on S d is given in Section 9.1 of [2].Since the intersection of every family of convex sets is also convex, for every set A ⊂ S d contained in an open hemisphere of S d there is the smallest convex set conv(A) containing Q. We call it the convex hull of A.Let C ⊂ S d be a convex body. Let Q ⊂ S d be a convex body or a hemisphere. We say that C touches Q from inside if C ⊂ Q and bd(C)∩bd(Q) = ∅. We say that C touches Q from outside if C ∩ Q = ∅ and int(C) ∩ int(Q) = ∅. In both cases, points of bd(C) ∩ bd(Q) are called points of touching. In the first case, if Q is a hemisphere, we also say that Q supports C, or supports C at t, provided t is a point of touching. If at every boundary point of C exactly one hemisphere supports C, we say that C is smooth. We call e ∈ C an extreme point of C if C \ {e} is convex.If hemispheres G and H of S d are different and not opposite, then L = G∩H is called a lune of S d . This notion is considered in many books and papers (for instance, see [12]).

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Reduced convex bodies in Euclidean space—A survey

Cited by 28 publications

References 35 publications

Width of spherical convex bodies

Width of spherical convex bodies

Vertex-facet assignments for polytopes

Spherical bodies of constant width

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