Pairs of non-intersecting random flats (Q1094735)

Let \(A_ r\), \(A_ s\), \(1\leq r+s\leq n-1\), be independent isotropic uniform random r- resp. s-flats meeting an n-dimensional convex body K. It is shown that the probability that the points realizing the distance of \(A_ r\) and \(A_ s\) belong to K is maximal if and only if K is a ball.