Spherical bodies of constant width

TL;DR

This paper studies spherical bodies of constant width on the sphere, proving their diameter equals the width, their strict convexity when the width is less than pi/2, and exploring conditions under which constant width and diameter bodies coincide.

Contribution

It introduces properties of spherical bodies of constant width, including their diameter and convexity, and investigates the relationship between constant width and constant diameter bodies.

Findings

Diameter of constant width bodies equals their width

Bodies with width less than pi/2 are strictly convex

Conditions for coincidence of constant width and constant diameter bodies

Abstract

The intersection $L$ of two different non-opposite hemispheres $G$ and $H$ of a $d$-dimensional 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 %spherical convex body $C \subset S^d$ we define ${\rm width}_G(C)$ as the thickness of the narrowest lune or lunes of the form $G \cap H$ containing $C$. If ${\rm width}_G(C) =w$ 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 are checking when spherical bodies of constant width and constant diameter coincide.