An Euler relation for valuations on polytopes,
Advances in Mathematics, 147 (1999), 1-34.
A locally finite point set (such as the set 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 Möbius inversion. In
this context a new family of inversion relations is obtained, thereby
generalizing classical relations of Euler, Dehn-Sommerville, and
Macdonald.
A pdf file for this paper is available
here.
Some special cases of the results in this paper are
considered from a more elementary point of view in
Free polygon enumeration and the area of an integral polygon.
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