Minimal-Volume Equiangular Hyperbolic $4$-Polytopes

Abstract

We study finite-volume convex hyperbolic $4$-polytopes whose dihedral angles are all equal to a fixed strictly acute angle $\alpha$. Put ${\alpha }_0=\arccos (1/3)$ and ${\alpha }_1=\arccos (1/4)$. We first show that the class is empty for $\alpha <{\alpha }_0$. For ${\alpha }_0\le \alpha <{\alpha }_1$, the unique polytope of minimum volume is the regular hyperbolic $4$-simplex with dihedral angle $\alpha$. As $\alpha$ increases to ${\alpha }_1$, this simplex degenerates to a Euclidean one and no hyperbolic simplex exists at ${\alpha }_1$. For ${\alpha }_1\le \alpha <\pi /2$, 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.