Energy minimization, periodic sets and spherical designs

TL;DR

This paper investigates conditions under which certain periodic point arrangements in Euclidean space minimize energy for various potentials, providing local optimality results and supporting a conjecture about the universal optimality of specific lattices.

Contribution

It establishes sufficient conditions for local energy minimization of periodic sets and proves a local version of a conjecture on the universal optimality of key lattices.

Findings

Proves local energy minimality for specific lattices.

Supports the conjecture that certain lattices are globally universally optimal.

Provides conditions applicable to a broad class of potential functions.

Abstract

We study energy minimization for pair potentials among periodic sets in Euclidean spaces. We derive some sufficient conditions under which a point lattice locally minimizes the energy associated to a large class of potential functions. This allows in particular to prove a local version of Cohn and Kumar's conjecture that $\mathsf{A}_2$, $\mathsf{D}_4$, $\mathsf{E}_8$ and the Leech lattice are globally universally optimal, regarding energy minimization, and among periodic sets of fixed point density.