Compact Riemannian manifold with maximum average distance

Question

Given a compact connected Riemannian $2$-manifold $M$ with positive curvature (thus by Gauß-Bonnet, $M$ is diffeomorphic to a 2-sphere) and diameter 1, what is the supremum (as $M$ varies over all such manifolds) of the average distance of two random points on $M$, i. e. the expectation value of the distance between two indepenently chosen randomly chosen points (distributed according to the volume form on $M$)?

A simple symmetry argument shows that this average distance is $\frac{1}{2}$ when $M$ is $S^2$ with its uniform metric. But is there any way to do better?

Answer 1

Likely the doubling of the disc is the best. Who can do better?

At least, if the diameter of the doubling is 1 then the average distance is bigger than $\tfrac12$. This shows that the optimal surface is not the round sphere (in particular, the symmetry is not maximal). In this case, the chances to prove that the doubling of the disc (or any other candidate) is optimal are nearly 0.

For example, the is a closely related problem of Alexandrov to maximize area among all the positively curved surfaces with diameter 1. The same picture --- round sphere is not the answer, the doubling of the disc is likely the best and the problem is open for more than 50 years.