An Euler Relation for Valuations on Polytopes

Abstract A locally finite point set (such as the set Z n of integral points) gives rise to a lattice of polytopes in Euclidean space taking vertices from the given point set. We develop the combinatorial structure of this polytope lattice and derive Euler-type relations for valuations on polytopes using the language of Mobius inversion. In this context a new family of inversion relations is obtained, thereby generalizing classical relations of Euler, Dehn–Sommerville, and Macdonald.

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