Local crystallization in three-dimensional quantization

Abstract

A long-standing conjecture on the quantization problem due to Gersho, which remains open in dimensions higher than two, posits that optimal quantizers crystallize into a lattice structure; in dimension three, the natural candidate is the body-centered cubic (BCC) lattice, known to be optimal among other lattices. In this paper, we establish the asymptotic local optimality of the BCC lattice under small periodic perturbations, extending its known stability beyond the class of lattice configurations. Our approach connects the discrete geometry of Voronoi tessellations with a continuum variational problem, showing that the rescaled quadratic quantization energies converge to a continuum functional describing locally deformed BCC lattices. A central ingredient is a geometric and algebraic framework that combines the stability of the BCC Voronoi structure, a discrete calculus on its adjacency graph and an analysis of the lattice symmetries via representation theory, allowing us to establish the positivity of the second variation at the BCC configuration and thereby prove the local minimality of the BCC lattice for the limiting energy.