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Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2018 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 20, no 9, p. 2181-2208Article in journal (Refereed) Published
Abstract [en]

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).

Place, publisher, year, edition, pages
European Mathematical Society Publishing House, 2018. Vol. 20, no 9, p. 2181-2208
Keywords [en]
Combinatorial positivity, Discrete Hadwiger theorem, Ehrhart polynomials, H*-vectors, Multivariate reciprocity, Translation-invariant valuations
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-238401DOI: 10.4171/JEMS/809ISI: 000440011500004Scopus ID: 2-s2.0-85052013303OAI: oai:DiVA.org:kth-238401DiVA, id: diva2:1259906
Note

QC 20181031

Available from: 2018-10-31 Created: 2018-10-31 Last updated: 2018-10-31Bibliographically approved

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Jochemko, Katharina

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