On the Riesz energy of measures

Abstract

Representations for the Riesz kernel |x−y| −s (s>0) are presented, which lead to new interpretations of the energy of measures. It is shown that the surface measure on the unit sphere in R d solves a minimal energy problem independent of s (but intimately related to Riesz s-energy) and that n points on the unit circle with minimal discrete Riesz energy are nth roots of unity, unique up to rotation. Moreover, the energy of signed measures is estimated in terms of their discrepancy.