Asymptotics of discrete Riesz $d$-polarization on subsets of $d$-dimensional manifolds

Abstract

We prove a conjecture of T. Erdélyi and E.B. Saff, concerning the form of the dominant term (as $N\to \infty$) of the $N$-point Riesz $d$-polarization constant for an infinite compact subset $A$ of a $d$-dimensional $C^1$-manifold embedded in ${\mathbb{R}}^m$ ($d\le m$). Moreover, if we assume further that the $d$-dimensional Hausdorff measure of $A$ is positive, we show that any asymptotically optimal sequence of $N$-point configurations for the $N$-point $d$-polarization problem on $A$ is asymptotically uniformly distributed with respect to ${\mathcal{H}}_d{|}_A$.

arXiv commentaccepted for publication in Potential Analysis

References in corpus (1)

• Analytic and Probabilistic Problems in Discrete Geometry