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On the volume of hyperbolic polyhedra. (English) Zbl 0664.51012
For ordinary orthoschemes in three-dimensional hyperbolic space Lobachevsky derived a volume formula in terms of the dihedral angles. In this article we study the volume problem for a more general class of hyperbolic polytopes, the so-called complete orthoschemes. These are orthoschemes truncated by at most two hyperplanes. The Coxeter polytopes (all dihedral angles are of the form \(\pi\) /p, \(p\in {\mathbb{N}}\), \(p\geq 2)\) among them arise as fundamental polytopes of a particular class of hyperbolic Coxeter groups and were classified by Im Hof in 1983. By generalizing the integration method of Coxeter and Böhm we derive explicit volume formulas for complete orthoschemes of dimension three; they are identical (up to a small modification in one particular case) to Lobachevsky’s formula for classical orthoschemes.
Reviewer: R.Kellerhals

MSC:
51M10 Hyperbolic and elliptic geometries (general) and generalizations
51M25 Length, area and volume in real or complex geometry
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References:
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