A maximal energy pointset configuration problem

TL;DR

This paper investigates a maximal energy pointset configuration problem with geometric constraints, explores its density limit, and develops a computational method for solving related problems, with applications to spheres and balls.

Contribution

It introduces a new extremal pointset configuration problem, establishes existence of solutions, and proposes a novel diffusion-generated computational method.

Findings

Solutions exist for the pointset and density problems.

Exact solutions are obtained for the $d$-dimensional ball and sphere.

The proposed method is proven to be monotonically increasing and effective for examples.

Abstract

We consider the extremal pointset configuration problem of maximizing a kernel-based energy subject to the geometric constraints that the points are contained in a fixed set, the pairwise distances are bounded below, and that every closed ball of fixed radius contains at least one point. We also formulate an extremal density problem, whose solution provides an upper bound for the pointset configuration problem in the limit as the number of points tends to infinity. Existence of solutions to both problems is established and the relationship between the parameters in the two problems is studied. Several examples are studied in detail, including the density problem for the $d$-dimensional ball and sphere, where the solution can be computed exactly using rearrangement inequalities. We develop a computational method for the density problem that is very similar to the Merriman-Bence-Osher (MBO) diffusion-generated method. The method is proven to be increasing for all non-stationary iterations and is applied to study more examples.