On a strengthening of the Blaschke–Leichtweiss theorem

Abstract

The Blaschke–Leichtweiss theorem (Leichtweiss in Abh Math Semin Univ Hambg 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 \({\mathbb {S}}^d\), \(d\ge 2\).