On the volume of hyperbolic polyhedra

Abstract

where V1k is the volume Vol~_ 2 (Sj~) of the apex Sjk:= Sj c~ S~ to wjk. Schl~ifli proved this formula for spherical simplices. In 1936, H. Kneser gave a second, very skilful proof of(l) (see I l l ] and [4, Sect. 5.1]) for both the spherical and hyperbolic cases (up to a change of sign in the latter case). But even for a three-dimensional non-euclidean simplex, the integration of this Schl~li differential is practically impossible. In fact, the most basic objects in polyhedral geometry are orthoschemes (or orthogonal-simplices) first introduced by Schl~fli: An n-orthoscheme R is an n-simplex with vertices Po . . . . . Pn such that