Approximation of spherical convex bodies of constant width $\pi/2$

TL;DR

This paper extends approximation results for spherical convex bodies of constant width to the critical case where the width is exactly a/2, showing such bodies can be approximated by spherical polytopes.

Contribution

It provides the first approximation of spherical bodies of constant width a/2 by spherical polytopes with arbitrary precision.

Findings

Any spherical convex body of width a/2 can be approximated by spherical polytopes.

The approximation can be made arbitrarily close in Hausdorff distance.

This completes the understanding of approximation for all constant widths on the sphere.

Abstract

Let $C\subset \mathbb{S}^2$ be a spherical convex body of constant width $\tau$. It is known that (i) if $\tau<\pi/2$ then for any $\varepsilon>0$ there exists a spherical convex body $C_\varepsilon$ of constant width $\tau$ whose boundary consists only of arcs of circles of radius $\tau$ such that the Hausdorff distance between $C$ and $C_\varepsilon$ is at most $\varepsilon$; (ii) if $\tau>\pi/2$ then for any $\varepsilon>0$ there exists a spherical convex body $C_\varepsilon$ of constant width $\tau$ whose boundary consists only of arcs of circles of radius $\tau-\frac{\pi}{2}$ and great circle arcs such that the Hausdorff distance between $C$ and $C_\varepsilon$ is at most $\varepsilon$. In this paper, we present an approximation of the remaining case $\tau=\pi/2$, that is, if $\tau=\pi/2$ then for any $\varepsilon>0$ there exists a spherical polytope $\mathcal{P}_\varepsilon$ of constant width $\pi/2$ such that the Hausdorff distance between $C$ and $\mathcal{P}_\varepsilon$ is at most $\varepsilon$.