Optimality of the BCC lattice among 3D lattice quantizers

We study the problem of the best approximation (w.r.t. the optimal transport metric $W_p$, $p \in [1,\infty)$) of the three-dimensional Lebesgue measure on a domain of volume 1 by a discrete probability measure supported on a Bravais lattice. This problem is known as the quantization problem and it arises in numerical integration, electrical engineering, discrete geometry and materials science. As the lattice size tends to zero and the cardinality of the support of the discrete measure tends to infinity, this reduces to the following geometric optimization problem: Amongst three-dimensional Bravais lattices with determinant $1$, which lattice minimises the $p$-th moment of the lattice Voronoi cell? Barnes and Sloane (1983) proved that the BCC lattice is optimal for the case $p=2$, which is known as the vector quantization problem in the signal compression literature. Bambah (1954) proved that the BCC lattice is optimal for the limiting case $p=\infty$, which is the sphere covering problem. Based on these results, Graf and Luschgy (2000) conjectured that the BCC lattice is optimal for all $p \in [1,\infty)$. Three dimensions is the critical dimension in the sense that in two dimensions the optimal lattice is independent of $p$, whereas in four dimensions it varies with $p$. In this paper we give support in favour of Graf and Luschgy's conjecture by proving that the BCC lattice is a local minimiser for all $p \in [1,\infty)$ and a global minimiser for $p$ near 2. While the conjecture asserts that the global minimiser is independent of $p$, we prove a bifurcation theorem which shows that the set of critical points depends on $p$ and that $p=2$ is a distinguished case. Our proofs combine results from discrete geometry by Barnes and Sloane (1983) and Conway and Sloane (1992) with optimal transport theory, the calculus of variations, bifurcation theory, and some delicate positivity estimates.