Three-point bounds for energy minimization

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 R P 2 \mathbb {R}\mathbb {P}^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 R P 2 \mathbb {R}\mathbb {P}^2 . We also give a new derivation of semidefinite programming bounds and present several surprising conjectures about them.