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Valuations and Tensor Weights on Polytopes

Part of: Polytopes and polyhedra

Published online by Cambridge University Press:  21 December 2009

Peter McMullen
Peter McMullen
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT.
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Abstract

Let be a finite-dimensional vector space over a square-root closed ordered field (this restriction permits an inner product with corresponding norm to be imposed on ). Many properties of the family :=() of convex polytopes in can be expressed in terms of valuations (or finitely additive measures). Valuations such as volume, surface area and the Euler characteristic are translation invariant, but others, such as the moment vector and inertia tensor, display a polynomial behaviour under translation. The common framework for such valuations is the polytope (or Minkowski) ring Π:=Π(), and its quotients under various powers of the ideal T of Π which is naturally associated with translations. A central result in the theory is that, in all but one trivial respect, the ring Π/T is actually a graded algebra over . Unfortunately, while the quotients Π/Tk+1 are still graded rings for k > 1, they now only possess a rational algebra structure; to obtain an algebra over , some (weak) continuity assumptions have to be made, although these can be achieved algebraically, by factoring out a further ideal A, the algebra ideal.

MSC classification

Secondary: 52B05: Combinatorial properties (number of faces, shortest paths, etc.) 52B45: Dissections and valuations (Hilbert's third problem, etc.)
Type
Research Article
Information
Mathematika , Volume 53 , Issue 1 , June 2006 , pp. 1 - 47
Copyright
Copyright © University College London 2006

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