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The optimal four-dimensional lattice quantizer and bounds from Voronoi facets

  • Tianle Liu
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Abstract

We prove that the root lattice D4 uniquely minimizes the normalized second moment among four-dimensional lattices, up to similarity. The computer-assisted proof combines the classification of secondary cones with integration over Voronoi cells and weighted sums of squares. In dimensions at least two, comparison with circular cones gives a lower bound in terms of the number of Voronoi facets, improving the sphere bound and recovering the hexagonal value in dimension two. We prove Ajjanagadde's circular-cone concavity conjecture and deduce the lattice case of his conjectured quantization bound.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026optimal,
  title = {{The optimal four-dimensional lattice quantizer and bounds from Voronoi facets}},
  author = {Tianle Liu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00100},
  eprint = {2611.00100},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00100},
}
RIS
TY  - UNPB
TI  - The optimal four-dimensional lattice quantizer and bounds from Voronoi facets
AU  - Tianle Liu
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00100
UR  - https://arxiv.org/abs/2611.00100
AB  - We prove that the root lattice D_4 uniquely minimizes the normalized second moment among four-dimensional lattices, up to similarity. The computer-assisted proof combines the classification of secondary cones with integration over Voronoi cells and weighted sums of squares. In dimensions at least two, comparison with circular cones gives a lower bound in terms of the number of Voronoi facets, improving the sphere bound and recovering the hexagonal value in dimension two. We prove Ajjanagadde's circular-cone concavity conjecture and deduce the lattice case of his conjectured quantization bound.
ER  -