Daniel Klain - An Euler relation for valuations on polytopes

A general Euler-type relation is derived for valuations on polytopes, leading in turn to formulas for valuations on a polytope P in terms of the enumeration of polytopes contained in P that are free with respect to a given locally finite set of points in Euclidean space. Special cases include Euler relations for volume, surface area, the Euler characteristic, and integer lattice point enumeration, as well as the Dehn-Sommerville equations and Macdonald's relation for compact simplicial manifolds.