2017/07/12 by Douglas P. Hardin, Alexander Reznikov, Hardin, D. P. +5
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.1707.03926
openalex publication_date 2017/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate separation properties of N-point configurations that minimize discrete Riesz s-energy on a compact set A⊂ ℝp. When A is a smooth (p-1)-dimensional manifold without boundary and s∈ [p-2, p-1), we prove that the order of separation (as N→ ∞) is the best possible. The same conclusions hold for the points that are a fixed positive distance from the boundary of A whenever A is any p-dimensional set. These estimates extend a result of Dahlberg for certain smooth (p-1)-dimensional surfaces when s=p-2 (the harmonic case). Furthermore, we obtain the same separation results for `greedy' s-energy points. We deduce our results from an upper regularity property of the s-equilibrium measure (i.e., the measure that solves the continuous minimal Riesz s-energy problem), and we show that this property holds under a local smoothness assumption on the set A.