arXiv:2611.00100 · v2
Optimality of the $D_4$ lattice quantizer
摘要
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.
作者备注：Simplified the proof of four-dimensional optimality, with explicit polynomial certificates now included in the paper. Main results unchanged