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Non-linear angle-sum relations for polyhedral cones and polytopes

Published online by Cambridge University Press:  24 October 2008

P. McMullen
P. McMullen
Affiliation:
University College, London
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Abstract

It is shown that the internal and external angles at the faces of a polyhedral cone satisfy various bilinear relations. The first two of these are related to the Gauss–Bonnet and Steiner parallel formulae for spherical polytopes, while the third is completely new. However, the proofs are basically combinatorial in nature, rather than differential geometric, as in the more classical treatments. These relations lead to inversion formulae, analogous to Euler-type relations, for certain functions defined on polytopes and polyhedral cones. As a result, various new relations involving quermassintegrals and Grassmann angles are found; there is also an application to lattice polytopes.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78 , Issue 2 , September 1975 , pp. 247 - 261
Copyright
Copyright © Cambridge Philosophical Society 1975

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