Random spheres in a convex body (Q1119256)

In d-dimensional Euclidean space \(E^ d(d\geq 2)\) a convex body k is given and \(m+1\), \(2\leq m\leq d\), points \(X_ 0,...,X_ m\) are chosen independently and uniformly at random in K. The probability p(K; m-1) that the (m-1)-dimensional circumsphere \(C_{m-1}\) of \(X_ 0,...,X_ m\) is entirely contained in K is determined for \(m=d\) and some extremal properties are given. For instance, p(K; d-1) is maximal if and only if K is a ball. For the case where K is the d-dimensional unit ball, the investigations are extended to m, \(2\leq m\leq d\).