Valuations on lattice polytopes

• Autores: Peter McMullen
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 220, Nº 1, 2009, págs. 303-323
• Idioma: inglés
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• Resumen

  • Let be a lattice (that is, a -module of finite rank), and let denote the family of convex polytopes with vertices in ; here, convexity refers to the underlying rational vector space . In this paper it is shown that any valuation on satisfies the inclusion�exclusion principle, in the strong sense that appropriate extension properties of the valuation hold. Indeed, the core result is that the class of a lattice polytope in the abstract group for valuations on can be identified with its characteristic function in . In fact, the same arguments are shown to apply to , when is a module of finite rank over an ordered ring, and more generally to appropriate families of (not necessarily bounded) polyhedra