Spherical geometry -- a survey on width and thickness of convex bodies

TL;DR

This survey explores the geometry of convex bodies on the sphere, focusing on width, thickness, and related properties, and draws parallels with Euclidean convex geometry using tools like hemispheres and lunes.

Contribution

It provides a comprehensive overview of spherical convex bodies, emphasizing the roles of width, thickness, and constant width, and connects these concepts with classical Euclidean notions.

Findings

Characterization of reduced spherical convex bodies

Properties of spherical bodies of constant width

Relations between width, diameter, and area on the sphere

Abstract

We present a survey article about the geometry of convex bodies on the $d$-dimensional sphere $S^d$. We concentrate on the results based on the notion of the width of a convex body $C \subset S^d$ determined by a supporting hemisphere of $C$. Important tools are the lunes containing $C$. The supporting hemispheres take over the role of the supporting half-spaces of a convex body in Euclidean space, and lunes the role of strips. Also essential is the notion of thickness of $C$, i.e, its minimum width. In particular, we describe properties of reduced spherical convex bodies and spherical bodies of constant width. The last notion coincides with the notions of complete bodies and bodies of constant diameter on $S^d$. The reminded and commented here results concern mostly the width, thickness, diameter, perimeter, area and extreme points of spherical convex bodies, reduced bodies and bodies of constant width.