Non-linear angle-sum relations for polyhedral cones and polytopes
- 1 September 1975
- journal article
- research article
- Published by Cambridge University Press (CUP) in Mathematical Proceedings of the Cambridge Philosophical Society
- Vol. 78 (2) , 247-261
- https://doi.org/10.1017/s0305004100051665
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.This publication has 19 references indexed in Scilit:
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