Mean distance between random points on the boundary of a convex body

Consider a convex figure $K$ on the plane. Let $\theta(K)$ denote the mean distance between two random points independently and uniformly selected on the boundary of $K$. The main result of the paper is that among all convex shapes of a fixed perimeter, the maximum value of $\theta(K)$ is reached at the circle and only at it. The continuity of $\theta(K)$ in the Hausdorff metric is also proved.