Polarization Optimality of Equally Spaced Points on the Circle for Discrete Potentials

We prove a conjecture of Ambrus, Ball and Erdélyi that equally spaced points maximize the minimum of discrete potentials on the unit circle whenever the potential is of the form $$\begin{aligned} \sum _{k=1}^n f(d(z,z_k)), \end{aligned}$$ ∑ k = 1 n f ( d ( z , z k ) ) , where $f:0,\pi ]\rightarrow 0,\infty ]$ f : 0 , π ] → 0 , ∞ ] is non-increasing and convex and $d(z,w)$ d ( z , w ) denotes the geodesic distance between z and w on the circle.