Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem

Abstract

We introduce the notion of combinatorial positivity of translation-invariant valuations on convex polytopes that extends the nonnegativity of Ehrhart $h^{\ast }$-vectors. We give a surprisingly simple characterization of combinatorially positive valuations that implies Stanley’s nonnegativity and monotonicity of $h^{\ast }$-vectors and generalizes work of Beck et al. (2010) from solid-angle polynomials to all translation-invariant simple valuations. For general polytopes, this yields a new characterization of the volume as the unique combinatorially positive valuation up to scaling. For lattice polytopes our results extend work of Betke–Kneser (1985) and give a discrete Hadwiger theorem: There is essentially a unique combinatorially-positive basis for the space of lattice-invariant valuations. As byproducts, we prove a multivariate Ehrhart–Macdonald reciprocity and we show universality of weight valuations studied in Beck et al. (2010).