The Mean Distance Between Random Points on the Boundary of a Convex Figure

We consider a convex figure K in the plane. Let θ(K) denote the average distance between two random points independently and uniformly chosen on the boundary of K. The main result of the note is that among all convex figures with a fixed perimeter, the maximum value of θ(K) is achieved by the circle and only by the circle. The continuity of θ(K) in the Hausdorff metric is also proven.