Optimal discrete measures for Riesz potentials

For weighted Riesz potentials of the form K(x,y)=w(x,y)/K(x,y)=w(x,y)/ |x−y|s|x-y|^s, we investigate NN-point configurations x1,x2,…,xNx_1,x_2, \ldots , x_N on a dd-dimensional compact subset AA of Rp\mathbb {R}^p for which the minimum of ∑j=1NK(x,xj)\sum _{j=1}^NK(x,x_j) on AA is maximal. Such quantities are called NN-point Riesz ss-polarization (or Chebyshev) constants. For s⩾ds\geqslant d, we obtain the dominant term as N→∞N\to \infty of such constants for a class of dd-rectifiable subsets of Rp\mathbb {R}^p. This class includes compact subsets of dd-dimensional C1C^1 manifolds whose boundary relative to the manifold has dd-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 NN-point configurations for weighted ss-polarization as N→∞N\to \infty.