On the Distribution of Weighted Extremal Points on a Surface in $$\mathbb{R}^d ,d \geqslant 3$$

Abstract

Consider the unit measure $$\mu _{F_n }$$ associating the mass $${{1{\kern 1pt} } \mathord{\left/ {\vphantom {{1{\kern 1pt} } n}} \right. \kern-\nulldelimiterspace} n}$$ with n points on a smooth surface in $$\mathbb{R}^d ,d \geqslant 3$$ , minimizing discrete energy under the influence of an external field $$f$$ . We call such points weighted extremal points. How well do the $$\mu _{F_n }$$ 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 $$\mu _{F_n }$$ and $$\mu _f$$ , for the discrete energy of $$\mu _{F_n }$$ and for the discrepancy $$\left| {\mu _{F_n } \left( B \right) - \mu _f \left( B \right)} \right|$$ , 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].