Self-dual Wulff shapes and spherical convex bodies of constant width ${\pi}/{2}$

Han, Huhe; Nishimura, Takashi

doi:10.2969/jmsj/06941475

Cited by 22 publications

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References 7 publications

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“…If for every hemisphere supporting a convex body W ⊂ S d the width of W determined by K is the same, we say that W is a body of constant width Vol. 92 (2018) Spherical bodies of constant width 633 (see [6] and for an application also [5]). In particular, spherical balls of radius smaller than π 2 are bodies of constant width.…”

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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Section: Spherical Bodies Of Constant Widthmentioning

confidence: 99%

Spherical bodies of constant width

Lassak

Musielak

2018

Aequat. Math.

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“…As a consequence of these facts, the above problem remains now open only for non-smooth bodies of constant diameter below π 2 . By the way, in [3], [4] and [6] spherical bodies of constant width and constant diameter π 2 are applied for recognizing if a Wullf shape is self-dual.…”

Section: Introductionmentioning

confidence: 99%

When is a spherical body of constant diameter of constant width?

Lassak

2020

Aequat. Math.

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“…By this fact and proposition 7, considering inside the sphere S n+1 seems to derive a reasonable situation. Moreover, notice that, by [24], the selfdual Wulff shapes are strongly related to the angle π/2 (see Subsection 3.6). Thus, the angle π/2 may be considered as a significant number for studying Wulff shapes, although in R n+1 there are no such significant real numbers for studying Wulff shapes.…”

Section: Propositionmentioning

confidence: 98%

“…Thus, Question 7 is the characterization question of a self-dual Wulff shape. A complete answer to this question with a rigorous proof and with many examples, which is summarized in Subsection 3.6, is obtained in [24].…”

Section: Questionmentioning

confidence: 99%

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Spherical method for studying Wulff shapes and related topics

Han¹,

Nishimura²

Singularities in Generic Geometry

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This is a survey article on the spherical method for studying Wulff shapes and related topics. The spherical method, which seems less common, is a powerful tool to study Wulff shapes and their related topics. It is verified how powerful the spherical method is by various results which seem difficult to be obtained without using the method. In this survey, the spherical method is explained in detail, and results obtained by using the spherical method until April 2016 are explained as well.A. Dinghas (1944, [14]) gave a formal proof. J. Taylor (1978, [69]) gave a precise proof for very general surface energies and a very general class of set for which the surface energy is defined by using geometric measure theory. B. Dacorogna and C. E. Pfister (1992, [13]) gave an analytic proof when n = 1. I. Fonseca (1991, [16]) and I. Fonseca and S. Müller (1991, [17]) gave a simpler proof for arbitrary dimensions.Let Γ γ be the boundary of the convex hull of inv(graph(γ)). Then, the following is known.Proposition 1 ([69, 23]). Let γ 1 , γ 2 be two elements of C 0 (S n , R + ) such that Γ γ1 = Γ γ2 . Then, the equality W γ1 = W γ2 holds.By Proposition 1, counterexamples for Question 3 are easily constructed (see Figure 3).As the next question, the following question naturally arises.

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Self-dual Wulff shapes and spherical convex bodies of constant width ${\pi}/{2}$

Cited by 22 publications

References 7 publications

Spherical bodies of constant width

Spherical bodies of constant width

When is a spherical body of constant diameter of constant width?

Spherical method for studying Wulff shapes and related topics

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