An Euler relation for valuations on polytopes
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.
scientific article; zbMATH DE number 1389125
scientific article