Geodesic
Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem
Journal of the European Mathematical Society, Tome 20 (2018) no. 9, pp. 2181-2208

Voir la notice de l'article provenant de la source EMS Press

We introduce the notion of combinatorial positivity of translation-invariant valuations on convex polytopes that extends the nonnegativity of Ehrhart h∗-vectors. We give a surprisingly simple characterization of combinatorially positive valuations that implies Stanley’s nonnegativity and monotonicity of h∗-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).
DOI : 10.4171/jems/809
Classification : 52-XX, 05-XX
Keywords: Ehrhart polynomials, h∗-vectors, combinatorial positivity, translation-invariant valuations, discrete Hadwiger theorem, multivariate reciprocity 
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     author = {Katharina Jochemko and Raman Sanyal},
     title = {Combinatorial positivity of translation-invariant valuations and a discrete {Hadwiger} theorem},
     journal = {Journal of the European Mathematical Society},
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Katharina Jochemko; Raman Sanyal. Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem. Journal of the European Mathematical Society, Tome 20 (2018) no. 9, pp. 2181-2208. doi: 10.4171/jems/809

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