Covering and separation of Chebyshev points for non-integrable Riesz potentials

For Riesz s-potentials K(x,y)=|x−y|−s, s>0, we investigate separation and covering properties of N-point configurations ωN∗={x1,…,xN} on a d-dimensional compact set A⊂Rℓ for which the minimum of ∑j=1NK(x,xj) is maximal. Such configurations are called N-point optimal Riesz s-polarization (or Chebyshev) configurations. For a large class of d-dimensional sets A we show that for s>d the configurations ωN∗ have the optimal order of covering. Furthermore, for these sets we investigate the asymptotics as N→∞ of the best covering constant. For these purposes we compare best-covering configurations with optimal Rieszs-polarization configurations and determine the sth root asymptotic behavior (as s→∞) of the maximal s-polarization constants. In addition, we introduce the notion of “weak separation” for point configurations and prove this property for optimal Riesz s-polarization configurations on A for s>dim(A), and for d−1⩽s