Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem
Katharina Jochemko
KTH - Royal Institute of Technology, Stockholm, SwedenRaman Sanyal
J. W. Goethe-Universität, Frankfurt, Germany
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Abstract
We introduce the notion of combinatorial positivity of translation-invariant valuations on convex polytopes that extends the nonnegativity of Ehrhart -vectors. We give a surprisingly simple characterization of combinatorially positive valuations that implies Stanley’s nonnegativity and monotonicity of -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).
Cite this article
Katharina Jochemko, Raman Sanyal, Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem. J. Eur. Math. Soc. 20 (2018), no. 9, pp. 2181–2208
DOI 10.4171/JEMS/809