An Euler relation for valuations on polytopes
type: Item
instance of: scholarly article
title: An Euler relation for valuations on polytopes
zbMATH Open document ID: 0944.52007
author: Daniel A. Klain
published in: Advances in Mathematics
publication date: 2000-09-11T00:00:00Z
review text: A notion common to both geometric and combinatorial settings is that of valuation which serves as both a generalization and a combinatorial analogue of the measures of classical analysis. They include important functionals, such as volume, surface area, and the Euler characteristic. The main result of this paper is a general family of Möbius inversion identities that characterize all valuations on polytopes. This result, theorem on free polytope inversion identity, generalizes classical relations of Euler, Dehn-Sommerville, and Macdonald, for functionals on convex polytopes. These identities are applied to some important examples of valuations on polytopes.
Mathematics Subject Classification ID: 52B45
zbMATH DE Number: 1389125
zbMATH Keywords: valuation
zbMATH Keywords: Möbius inversion identities
zbMATH Keywords: free polytope inversion identity
zbMATH Keywords: order ideals
zbMATH Keywords: induced valuations
zbMATH Keywords: polytope valuations
zbMATH Keywords: visible decomposition of boundary
zbMATH Keywords: measures
zbMATH Keywords: volume
zbMATH Keywords: surface area
zbMATH Keywords: Euler characteristic
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
OpenAlex ID: W2148061832
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DOI: 10.1006/AIMA.1999.1831
description: scientific article; zbMATH DE number 1389125