Asymptotics of greedy energy points

Abstract

For a symmetric kernel k : X × X → R ∪ { + ∞ } k:X\times X \rightarrow \mathbb {R}\cup \{+\infty \} on a locally compact metric space X X , we investigate the asymptotic behavior of greedy k k -energy points { a i } 1 ∞ \{a_{i}\}_{1}^{\infty } for a compact subset A ⊂ X A\subset X that are defined inductively by selecting a 1 ∈ A a_{1}\in A arbitrarily and a n + 1 a_{n+1} so that ∑ i = 1 n k ( a n + 1 , a i ) = inf x ∈ A ∑ i = 1 n k ( x , a i ) \sum _{i=1}^{n}k(a_{n+1},a_{i})=\inf _{x\in A}\sum _{i=1}^{n}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 ∑ i ≠ j N k ( a i , a j ) \sum _{i\neq j}^{N}k(a_{i},a_{j}) as N → ∞ N\rightarrow \infty that is asymptotically the same as E ( A , N ) := min { ∑ i ≠ j k ( x i , x j ) : x 1 , … , x N ∈ A } \mathcal {E}(A,N):=\min \{\sum _{i\neq j}k(x_{i},x_{j}):x_{1},\ldots ,x_{N}\in A\} ), and have asymptotic distribution equal to the equilibrium measure for A A . For the case of Riesz kernels k s ( x , y ) := | x − y | − s k_{s}(x,y):=|x-y|^{-s} , s > 0 s>0 , we show that if A A is a rectifiable Jordan arc or closed curve in R p \mathbb {R}^{p} and s > 1 s>1 , then greedy k s k_{s} -energy points are not asymptotically energy minimizing, in contrast to the case s > 1 s>1 . (In fact, we show that no sequence of points can be asymptotically energy minimizing for s > 1 s>1 .) Additional results are obtained for greedy k s k_{s} -energy points on a sphere, for greedy best-packing points (the case s = ∞ s=\infty ), and for weighted Riesz kernels. For greedy best-packing points we provide a simple counterexample to a conjecture attributed to L. Bos.