Polarization Problem on a Higher-Dimensional Sphere for a Simplex

We study the problem of maximizing the minimal value over the sphere S d - 1 ⊂ R d of the potential generated by a configuration of d + 1 points on S d - 1 (the maximal discrete polarization problem). The points interact via the potential given by a function f of the Euclidean distance squared, where f : [ 0 , 4 ] → ( - ∞ , ∞ ] is continuous (in the extended sense), decreasing on [0, 4], and finite and convex on (0, 4], with a concave or convex derivative f ′ . We prove that the configuration of the vertices of a regular d -simplex inscribed in S d - 1 is optimal. This result is new for d > 3 (certain special cases for d = 2 and d = 3 are also new). As a byproduct, we find a simpler proof for the known optimal covering property of the vertices of a regular d -simplex inscribed in S d - 1 .