Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 46, No. 2, pp. 609-614 (2005)
On the monotonicity of the volume of hyperbolic convex polyhedra
Károly BezdekDepartment of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Ab Canada T2N 1N4; e-mail: firstname.lastname@example.org
Abstract: We give a proof of the monotonicity of the volume of nonobtuse-angled compact convex polyhedra in terms of their dihedral angles. More exactly we prove the following. Let $P$ and $Q$ be nonobtuse-angled compact convex polydedra of the same simple combinatorial type in hyperbolic $3$-space. If each (inner) dihedral angle of $Q$ is at least as large as the corresponding (inner) dihedral angle of $P$, then the volume of $P$ is at least as large as the volume of $Q$. Moreover, we extend this result to nonobtuse-angled hyperbolic simplices of any dimension.
Keywords: hyperbolic nonobtuse-angled convex polyhedra, Koebe-Andreev- Thurston theorem, hyperbolic volume
Classification (MSC2000): 52A55, 52C29
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Electronic version published on: 18 Oct 2005. This page was last modified: 29 Dec 2008.