The volume of the Lambert cube in spherical space

Abstract

The Lambert cube Q(α, β, γ) is one of the simplest polyhedra. By definition, this is a combinatorial cube with dihedral angles α, β, and γ at three noncoplanar edges and with right angles at all other edges. The volume of the Lambert cube in hyperbolic space was obtained by R. Kellerhals (1989) in terms of the Lobachevskii function Λ(x). In the present paper, we find the volume of the Lambert cube in spherical space. It is expressed in terms of the function $$ \delta (\alpha ,\theta ) = \int_\theta ^{\pi /2} {\log (1 - \cos 2\alpha \cos 2\tau )} \frac{{d\tau }} {{\cos 2\tau }}, $$ which can be regarded as the spherical analog of the function $$ \Delta (\alpha ,\theta ) = \Lambda (\alpha + \theta ) - \Lambda (\alpha - \theta ). $$