A new look at the Blaschke-Leichtweiss theorem

2021. 04. 27. 16:00
Online, ZOOM
Bezdek Károly

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 $\S^2$ 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 $\S^2$, 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 short proof for 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 $\S^d$ for all $d\geq 2$. Also, it is shown that any minimum volume wide $r$-ball body is of constant width $r$ in $\S^d$, $d\geq 2$.