Optimal discrete measures for Riesz potentials

Abstract

For $s⩾d$, we obtain the leading term as $N\to \infty$ of the maximal weighted $N$-point Riesz $s$-polarization (or Chebyshev constant) for a certain class of $d$-rectifiable compact subsets of ${\mathbb{R}}^p$. This class includes compact subsets of $d$-dimensional $C^1$ manifolds whose boundary relative to the manifold has ${\mathcal{H}}_d$-measure zero, as well as finite unions of such sets when their pairwise intersections have ${\mathcal{H}}_d$-measure zero. We also explicitly find the weak${}^{\ast }$ limit distribution of asymptotically optimal $N$-point polarization configurations as $N\to \infty$.

References in corpus (4)

• Analytic and Probabilistic Problems in Discrete Geometry
• Extremal Polarization Configurations for Integrable Kernels
• The covering radius of randomly distributed points on a manifold
• A minimum principle for potentials with application to Chebyshev constants