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 $\epsilon >0$ there exists a spherical convex body $C_{\epsilon }$ of constant width $\tau$ whose boundary consists only of arcs of circles of radius $\tau$ such that the Hausdorff distance between $C$ and $C_{\epsilon }$ is at most $\epsilon$; (ii) if $\tau >\pi /2$ then for any $\epsilon >0$ there exists a spherical convex body $C_{\epsilon }$ 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 $C_{\epsilon }$ is at most $\epsilon$. In this paper, we present an approximation of the remaining case $\tau =\pi /2$, that is, if $\tau =\pi /2$ then for any $\epsilon >0$ there exists a spherical polytope ${\mathcal{P}}_{\epsilon }$ of constant width $\pi /2$ such that the Hausdorff distance between $C$ and ${\mathcal{P}}_{\epsilon }$ is at most $\epsilon$.

7 pages,2 figures