The mean distance between random points on the boundary of a convex figure (Q6657314)

In this paper, the expected distance of two random points is considered, where the points are chosen uniformly and independently on the boundary of a planar convex figure \(K\). The author shows an upper bound for the expectation, namely that among convex figures of fixed perimeter, the maximum of the expected distance is attained by the disc and only the disc. It is also shown that, unlike in the random model in which the points are chosen uniformly inside \(K\), this functional is continuous with respect to the Hausdorff metric.