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arXiv · 2611.00100

Optimality of the $D_4$ lattice quantizer

Tianle Liu
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Abstract

We prove that the root lattice $D_4$ uniquely minimizes the normalized second moment among four-dimensional lattices, up to similarity. The minimum is $13/(120\sqrt{2})$. The proof reduces the problem to polynomial inequalities on three secondary cones. We derive the second-moment formulas geometrically and give explicit polynomial certificates that can be checked by coefficient comparison. We also derive a lower bound for lattice quantizers in every dimension at least two from the number of Voronoi facets, using a comparison with circular cones and concavity in the solid angle. This bound strictly improves the sphere bound and recovers the hexagonal value in dimension two.

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Tianle Liu. 2026-11-03. Optimality of the $D_4$ lattice quantizer. https://arxiv.org/abs/2611.00100

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