Optimal discrete measures for Riesz potentials

Abstract:

For weighted Riesz potentials of the form $K(x,y)=w(x,y)/$ $|x-y|^s$, we investigate $N$-point configurations $x_1,x_2, \ldots , x_N$ on a $d$-dimensional compact subset $A$ of $\mathbb {R}^p$ for which the minimum of $\sum _{j=1}^NK(x,x_j)$ on $A$ is maximal. Such quantities are called $N$-point Riesz $s$-polarization (or Chebyshev) constants. For $s\geqslant d$, we obtain the dominant term as $N\to \infty$ of such constants for a class of $d$-rectifiable subsets of $\mathbb {R}^p$. This class includes compact subsets of $d$-dimensional $C^1$ manifolds whose boundary relative to the manifold has $d$-dimensional Hausdorff measure zero, as well as finite unions of such sets when their pairwise intersections have measure zero. We also explicitly determine the weak-star limit distribution of asymptotically optimal $N$-point configurations for weighted $s$-polarization as $N\to \infty$.