Optimal discrete measures for Riesz potentials

Abstract

For $s\geqslant d$, 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$^*$ limit distribution of asymptotically optimal $N$-point polarization configurations as $N\to \infty$.