Polarization optimality of equally spaced points on the circle for discrete potentials

We prove a conjecture of Ambrus, Ball and Erd{e}lyi that equally spaced points maximize the minimum of discrete potentials on the unit circle whenever the potential is of the form sum_{k=1}^n f(d(z,z_k)), where $f:[0,pi]to [0,infty]$ is non-increasing and strictly convex and $d(z,w)$ denotes the geodesic distance between $z$ and $w$ on the circle.