Polarization

Abstract

. We investigate the behavior of a greedy sequence on the sphere S d defined so that at each step the point that minimizes the Riesz s -energy is added to the existing set of points. We show that for 0 < s < d , the greedy sequence achieves optimal second-order behavior for the Riesz s -energy (up to constants). In order to obtain this result, we prove that the second-order term of the maximal polarization with Riesz s -kernels is of order N s/d in the same range 0 < s < d . Furthermore, using the Stolarsky principle relating the L 2 -discrepancy of a point set with the pairwise sum of distances (Riesz energy with s = − 1), we also obtain a simple upper bound on the L 2 -spherical cap discrepancy of the greedy sequence and give numerical examples that indicate that the true discrepancy is much lower.