On a strengthening of the Blaschke-Leichtweiss theorem

Abstract

The Blaschke-Leichtweiss theorem (Abh. Math. Sem. Univ. Hamburg 75: 257-284, 2005) states that the smallest area convex domain of constant width $w$ in the $2$-dimensional spherical space ${\mathbb{S}}^2$ is the spherical Reuleaux triangle for all $0<w\le \frac{\pi }{2}$. In this paper we extend this result to the family of wide $r$-disk domains of ${\mathbb{S}}^2$, where $0<r\le \frac{\pi }{2}$. Here a wide $r$-disk domain is an intersection of spherical disks of radius $r$ with centers contained in their intersection. This gives a new and elementary proof of the Blaschke-Leichtweiss theorem. Furthermore, we investigate the higher dimensional analogue of wide $r$-disk domains called wide $r$-ball bodies. In particular, we determine their minimum spherical width (resp., inradius) in the spherical $d$-space ${\mathbb{S}}^d$ for all $d\ge 2$. Also, it is shown that any minimum volume wide $r$-ball body is of constant width $r$ in ${\mathbb{S}}^d$, $d\ge 2$.

arXiv comment11 pages, 3 figures