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

For a compact right-angled polyhedron R in Lobachevskii space H(3), let vol(R) denote its volume and vert(R), the number of its vertices. Upper and lower bounds for vol(R) were recently obtained by Atkinson in terms of vert(R). In constructing a two-parameter family of polyhedra, we show that the asymptotic upper bound 5(v3)/8, where v(3) is the volume of the ideal regular tetrahedron in H(3), is a double limit point for the ratios vol(R)/ vert(R). Moreover, we improve the lower bound in the case vert(R) <= 56.