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PreprintCC BY 4.0arXiv

Minimal-Volume Equiangular Hyperbolic $4$-Polytopes

  • Andrey Egorov
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Abstract

We study finite-volume convex hyperbolic $4$-polytopes whose dihedral angles are all equal to a fixed strictly acute angle α. Put α0=arccos(1/3) and α1=arccos(1/4). We first show that the class is empty for α<α0. For α0≤α<α1, the unique polytope of minimum volume is the regular hyperbolic $4$-simplex with dihedral angle α. As α increases to α1, this simplex degenerates to a Euclidean one and no hyperbolic simplex exists at α1. For α1≤α<π/2, the unique minimum is the regular equiangular hyperbolic $4$-cube. The proof combines the hyperbolic Gram--Euler relation with the Davis--Okun theorem on the Charney--Davis inequality for flag triangulations of the $3$-sphere. The geometric step is a missing-face argument showing that the boundary of the dual of every nonsimplex in the class is flag.

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Cite

BibTeXRIS
BibTeX
@misc{egorov2026minimalvolume,
  title = {{Minimal-Volume Equiangular Hyperbolic $4$-Polytopes}},
  author = {Andrey Egorov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04258},
  eprint = {2609.04258},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04258},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Minimal-Volume Equiangular Hyperbolic $4$-Polytopes
AU  - Andrey Egorov
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.04258
UR  - https://arxiv.org/abs/2609.04258
AB  - We study finite-volume convex hyperbolic $4$-polytopes whose dihedral angles are all equal to a fixed strictly acute angle α. Put α_0=arccos(1/3) and α_1=arccos(1/4). We first show that the class is empty for α<α_0. For α_0≤α<α_1, the unique polytope of minimum volume is the regular hyperbolic $4$-simplex with dihedral angle α. As α increases to α_1, this simplex degenerates to a Euclidean one and no hyperbolic simplex exists at α_1. For α_1≤α<π/2, the unique minimum is the regular equiangular hyperbolic $4$-cube. The proof combines the hyperbolic Gram--Euler relation with the Davis--Okun theorem on the Charney--Davis inequality for flag triangulations of the $3$-sphere. The geometric step is a missing-face argument showing that the boundary of the dual of every nonsimplex in the class is flag.
ER  -