On a strengthening of the Blaschke-Leichtweiss theorem

TL;DR

This paper extends the Blaschke-Leichtweiss theorem to wide r-disk domains on the sphere, providing new proofs and analyzing higher-dimensional analogues called wide r-ball bodies, including their minimal width and volume properties.

Contribution

It generalizes the Blaschke-Leichtweiss theorem to wide r-disk domains and introduces the concept of wide r-ball bodies in higher dimensions with their properties.

Findings

Extended the theorem to wide r-disk domains on ${\f S}^2$.

Determined minimal spherical width and inradius of wide r-ball bodies in ${\bf S}^d$.

Proved that minimal volume wide r-ball bodies have constant width r.

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 $0<w\leq\frac{\pi}{2}$. In this paper we extend this result to the family of wide $r$-disk domains of ${\mathbb S}^2$, where $0<r\leq\frac{\pi}{2}$. 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 $d\geq 2$. Also, it is shown that any minimum volume wide $r$-ball body is of constant width $r$ in ${\mathbb S}^d$, $d\geq 2$.