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

TL;DR

This paper characterizes convex bodies of constant diameter on the sphere, showing they are also of constant width under certain smoothness or dimensional conditions.

Contribution

It provides a complete characterization of convex bodies with constant diameter as those with constant width on the sphere, under specific smoothness and dimension constraints.

Findings

Convex bodies of constant diameter are of constant width if smooth.

On the 2D sphere, constant diameter implies constant width.

The results hold for convex bodies with smooth boundaries or in two dimensions.

Abstract

{\bf Abstract.} Let $D$ be a convex body of diameter $\delta$, where $0 < \delta < \frac{\pi}{2}$, on the $d$-dimensional sphere. We prove that $D$ is of constant diameter $\delta$ if and only if it is of constant width $\delta$ in the following two cases. The first case is when $D$ is smooth. The second case is when $d=2$.