Advances in Mathematics 147, 134 (1999) An Euler Relation for Valuations on Polytopes

A locally finite point set (such as the set Zn of integral points) gives rise to a lat-tice 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 Mo bius inversion. In this context a new family of inversion relations is obtained, thereby generalizing classical relations of Euler, DehnSommerville, and Macdonald. 1999 Academic Press AMS Subject Classification: 52B45; 52B05, 52B20, 52B70, 52A38, 05E25 The essential link between convex geometry and combinatorial theory is the lattice structure of the collection of polyconvex sets; that is, the collec-tion of all finite unions of compact convex sets in Rn (also known as the convex ring). The present paper is inspired by the role of valuation theory in these two related contexts. Specifically, we consider the lattice of polytopes (ordered by inclusion) taking all vertices from a prescribed locally finite collection of points in Euclidean space. Much attention has been given to the specific example o