The hexagonal lattice is universally locally optimal

2025/11/05 by Leblé, Thomas
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Approximation and Integration #Mathematical Physics (math-ph) #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · doi:10.48550/arxiv.2511.03353

openalex publication_date 2025/11/05 · openalex created_date 2025/11/07 · openalex updated_date 2026/07/28

Abstract

We prove that the hexagonal lattice is a local minimizer, among all point configurations, of the interaction energy per unit volume for pair potentials that are completely monotonic functions of the square distance. This includes Gaussian interactions and power laws.

Citations

• Universally Optimal Periodic Configurations in the Plane
• Universal optimality of the E₈ and Leech lattices and interpolation formulas

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• Optimal and non-optimal lattices for non-completely monotone interaction potentials
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