Abstract

The intersection L of two different non-opposite hemispheres G and H of the d-dimensional 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 subset S^d we define mathrm{width}_G(C) as the thickness of the narrowest lune or lunes of the form G cap H containing C. If mathrm{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 < frac{pi }{2}, then C is strictly convex. Moreover, we check when spherical bodies of constant width and constant diameter coincide.

Highlights

  • Consider the unit sphere Sd in the (d + 1)-dimensional Euclidean space Ed+1 for d ≥ 2

  • By a (d − 1)-dimensional great sphere of Sd we mean the common part of Sd with any hyper-subspace of Ed+1

  • If for every hemisphere supporting a convex body W ⊂ Sd the width of W determined by K is the same, we say that W is a body of constant width

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Summary

Introduction

Consider the unit sphere Sd in the (d + 1)-dimensional Euclidean space Ed+1 for d ≥ 2. Set of points of a (d − 1)-dimensional great sphere of Sd which are at distances at most ρ from a fixed point. We call it the center of this ball. We define the thickness Δ(L) of a lune L = G ∩ H on Sd as the spherical distance of the centers of the (d − 1)-dimensional hemispheres G/H and H/G bounding L. We define the thickness Δ(C) of a spherical convex body C as the smallest width of C. By the relative interior of a convex set C ⊂ Sd we mean the interior of C with respect to the smallest sphere Sk ⊂ Sd that contains C

A few lemmas on spherical convex bodies
Spherical bodies of constant width
Constant width and constant diameter

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