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