Optimal logarithmic energy points on the unit sphere

Abstract

We study minimum energy point charges on the unit sphere S d in R d+1 , d ≥ 3, that interact according to the logarithmic potential log (1/r), where r is the Euclidean distance between points. Such optimal N-point configurations are uniformly distributed as N →∞. We quantify this result by estimating the spherical cap discrepancy of optimal energy configurations. The estimate is of order O (N- 1/(d+2) ). Essential is an improvement of the lower bound of the optimal logarithmic energy which yields the second term (1/d)(log N)/N in the asymptotical expansion of the optimal energy. Previously, this was known for the unit sphere in R 3 only. Furthermore, we present an upper bound for the error of integration for an equally-weighted numerical integration rule Q N with the N nodes forming an optimal logarithmic energy configuration. For polynomials p of degree at most n this bound is Cd(N 1 /d/n) -d/2 ||p||∞ as n/N 1/d → 0. For continuous functions f of S d satisfying a Lipschitz condition with constant C f the bound is (12dC f + C' d ||f||∞)O(N -1/(d+2) ) as N → ∞.