Characterizing the round sphere by mean distance

We discuss the measure theoretic metric invariants extent, rendezvous number and mean distance of a general compact metric space X and relate these to classical metric invariants such as diameter and radius. In the final section we focus attention to the category of Riemannian manifolds. The main result Of this paper is Theorem 4 stating that the round sphere S-1(n) of constant curvature 1 has maximal mean distance among Riemannian n-manifolds with Ricci curvature Ric >= n - 1, and that such a manifold is diffeomorphic to a sphere if the mean distance is close to pi/2.