Asymptotics of greedy energy points

For a symmetric kernel k: X x X → ℝ U {+∞} on a locally compact metric space X, we investigate the asymptotic behavior of greedy k-energy points {a i } ∞ 1 for a compact subset A C X that are defined inductively by selecting a 1 ∈ A arbitrarily and a n+1 so that Σ n i=1 k(a n+l1 ,a i ) = inf x∈A Σ n i =1 k(x,a i ). We give sufficient conditions under which these points (also known as Leja points) are asymptotically energy minimizing (i.e. have energy Σ N i≠j k(a i , aj) as N → ∞ that is asymptotically the same as e(A, N) := min{Σ i≠j k(x i , x j ): x 1 ,...,x N ∈ A}), and have asymptotic distribution equal to the equilibrium measure for A. For the case of Riesz kernels k s (x, y) := |x ― y| ―s , s > 0, we show that if A is a rectifiable Jordan arc or closed curve in RP and s > 1, then greedy k s -energy points are not asymptotically energy minimizing, in contrast to the case s 1.) Additional results are obtained for greedy k s -energy points on a sphere, for greedy best-packing points (the case s = ∞), and for weighted Riesz kernels. For greedy best-packing points we provide a simple counterexample to a conjecture attributed to L. Bos.