Two-sided bounds for the volume of right-angled hyperbolic polyhedra

TL;DR

This paper investigates bounds on the volume of compact right-angled hyperbolic polyhedra, showing asymptotic behavior and improving lower bounds for certain vertex counts, advancing understanding of hyperbolic polyhedral geometry.

Contribution

It introduces a 2-parameter family of polyhedra to analyze volume-vertex ratios, revealing asymptotic limits and refining lower bounds for specific vertex counts.

Findings

Asymptotic upper bound of 5v_3/8 for volume-to-vertex ratio

Double limit point for volume/vertex ratios at the asymptotic bound

Improved lower bounds for polyhedra with up to 56 vertices

Abstract

For a compact right-angled polyhedron $R$ in $\mathbb H^3$ denote by $\operatorname{vol} (R)$ the volume and by $\operatorname{vert} (R)$ the number of vertices. Upper and lower bounds for $\operatorname{vol} (R)$ in terms of $\operatorname{vert} (R)$ were obtained in \cite{A09}. Constructing a 2-parameter family of polyhedra, we show that the asymptotic upper bound $5 v_3 / 8$, where $v_3$ is the volume of the ideal regular tetrahedron in $\mathbb H^3$, is a double limit point for ratios $\operatorname{vol} (R) / \operatorname{vert} (R)$. Moreover, we improve the lower bound in the case $\operatorname{vert} (R) \leqslant 56$.