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

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 $\varepsilon$.