Three-point bounds for energy minimization

TL;DR

This paper develops three-point semidefinite programming bounds to establish universal optimality of certain seven-line configurations in RP^2, advancing understanding of energy minimization in spherical codes.

Contribution

It introduces new bounds for potential energy minimization and proves the universal optimality of specific seven-line configurations in RP^2, including the first non-distance-regular universal optimum.

Findings

Bounds are sharp for seven points in RP^2.

The seven lines form a universally optimal configuration.

The configuration is unique among known universal optima.

Abstract

Three-point semidefinite programming bounds are one of the most powerful known tools for bounding the size of spherical codes. In this paper, we use them to prove lower bounds for the potential energy of particles interacting via a pair potential function. We show that our bounds are sharp for seven points in RP^2. Specifically, we prove that the seven lines connecting opposite vertices of a cube and of its dual octahedron are universally optimal. (In other words, among all configurations of seven lines through the origin, this one minimizes energy for all potential functions that are completely monotonic functions of squared chordal distance.) This configuration is the only known universal optimum that is not distance regular, and the last remaining universal optimum in RP^2. We also give a new derivation of semidefinite programming bounds and present several surprising conjectures about them.