A new look at the Blaschke-Leichtweiss theorem

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 is the spherical Reuleaux triangle for all 0 < w ≤ π 2 . In this paper we extend this result to the family of wide r-disk domains of S, where 0 < r ≤ π 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 for all d ≥ 2. Also, it is shown that any minimum volume wide r-ball body is of constant width r in S, d ≥ 2.