四维球体填充问题与二十四胞体猜想

中文摘要 AI 辅助

在 $\mathbb{R}^4$ 中，单位球体填充的每个 Voronoi 胞腔的体积至少为 $8$，且仅 $D_4$ 构型达到该下界，因此二十四胞体猜想成立，并由此得到密度上界 $\Delta_4=\pi^2/16$。除二十四个接触点的情况外，所有接触数均已在此解决；该情况依赖于 de Laat、Leijenhorst 和 de Muinck Keizer 对最小角度为 $60^\circ$ 的二十四点码的分类。本文基于已发表数据，以精确算术重新验证了他们的证书，涵盖全部七个步骤，因此其计算中没有任何部分被盲目信任。

英文摘要

We prove that every Voronoi cell of a unit-ball packing of $\mathbb{R}^4$ 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.

发表机构

• Electro-Gravitational Space Propulsion Laboratory (EGSPL)（电引力空间推进实验室）
• National Taiwan University（台湾大学）
• Department of Mathematics, Bioinformatics and Computer Applications, Maulana Azad National Institute of Technology Bhopal（莫拉纳阿扎德国家技术研究所博帕尔数学、生物信息学和计算机应用系）
• Asian College of Teachers（亚洲教师学院）

机构由 AI 辅助整理，请以论文原文为准。