A sequence of discrete minimal energy configurations that does not converge in the weak-star topology

TL;DR

This paper constructs a specific example of a set and parameter where minimal energy configurations of points do not converge in the weak-star topology as the number of points increases.

Contribution

It provides a counterexample showing that minimal energy configurations can fail to have a weak-star limit, challenging assumptions about their asymptotic behavior.

Findings

Counterexample of non-converging energy configurations

Demonstrates limitations of weak-star convergence in energy minimization

Highlights need for refined understanding of asymptotic distributions

Abstract

We demonstrate a set A and a value of s for which the sequence of N-point discrete minimal Riesz s-energy configurations on A does not have an asymptotic distribution in the weak-star sense as N tends to infinity.