PreprintCC BY 4.0arXiv
The Sphere Packing Problem in Dimension 4 and the Twenty-Four-Cell Conjecture
Abstract
We prove that every Voronoi cell of a unit-ball packing of ℝ4 has volume at least $8$, with equality only at the D4 configuration, so that the twenty-four-cell conjecture holds and the density of a sphere packing in four dimensions is at most π2/16. The proof runs through the number of contacts of a cell. Up to twenty-two a covering estimate suffices; at twenty-three the cell is bounded through an exact volume identity inside a ball and a semidefinite certificate for one inequality between pair angles; at twenty-four the configuration is the root system, which we prove from the second level of the semidefinite hierarchy with its equality case, the positive kernel verified in exact arithmetic.
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BibTeX
@misc{bhattacharjee2026sphere,
title = {{The Sphere Packing Problem in Dimension 4 and the Twenty-Four-Cell Conjecture}},
author = {Deep Bhattacharjee and Ushashi Bhattacharya and Priyabrata Mandal and Shounak Bhattacharya},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25120},
eprint = {2609.25120},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25120},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Sphere Packing Problem in Dimension 4 and the Twenty-Four-Cell Conjecture
AU - Deep Bhattacharjee
AU - Ushashi Bhattacharya
AU - Priyabrata Mandal
AU - Shounak Bhattacharya
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.25120
UR - https://arxiv.org/abs/2609.25120
AB - We prove that every Voronoi cell of a unit-ball packing of ℝ^4 has volume at least $8$, with equality only at the D_4 configuration, so that the twenty-four-cell conjecture holds and the density of a sphere packing in four dimensions is at most π^2/16. The proof runs through the number of contacts of a cell. Up to twenty-two a covering estimate suffices; at twenty-three the cell is bounded through an exact volume identity inside a ball and a semidefinite certificate for one inequality between pair angles; at twenty-four the configuration is the root system, which we prove from the second level of the semidefinite hierarchy with its equality case, the positive kernel verified in exact arithmetic.
ER -