On the distribution of weighted extremal points on a surface in R-d, d >= 3

摘要

Consider the unit measure mu (Fn) associating the mass 1 with n points on a smooth surface in R-d, d greater than or equal to3, minimizing discrete energy under the influence of an external field f. We call such points weighted extremal points. How well do the mu (Fn) approximate the f-weighted equilibrium distribution mu (f) of the surface? We answer this question by presenting sharp estimates for the difference of the potentials of mu (Fn) and mu (f), for the discrete energy of mu (Fn) and for the discrepancy sup mu (Fn)(B)-mu (f)(B) , where the supremum is taken over a reasonable class of test sets B. In the unweighted case f=0, extremal points reduce to d-dimensional Fekete points, and, up to a logarithmic term, the presented discrepancy estimate solves a conjecture of J. Korevaar [13]. [References: 20]