Two-Sided Bounds for the Volume of Right-Angled Hyperbolic Polyhedra

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