The $h^{\ast }$-polynomials of locally anti-blocking lattice polytopes and their $\gamma$-positivity

Abstract

A lattice polytope $\mathcal{P}\subset {\mathbb{R}}^d$ is called a locally anti-blocking polytope if for any closed orthant ${\mathbb{R}}_{\epsilon }^d$ in ${\mathbb{R}}^d$, $\mathcal{P}\cap {\mathbb{R}}_{\epsilon }^d$ is unimodularly equivalent to an anti-blocking polytope by reflections of coordinate hyperplanes. In the present paper, we give a formula for the $h^{\ast }$-polynomials of locally anti-blocking lattice polytopes. In particular, we discuss the $\gamma$-positivity of the $h^{\ast }$-polynomials of locally anti-blocking reflexive polytopes.

19 pages, Explanations and references are added. Writings are improved

References in corpus (2)

• Many faces of symmetric edge polytopes
• Toric h-vectors and Chow Betti Numbers of Dual Hypersimplices

Cited by in corpus (7)

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• PQ-type adjacency polytopes of join graphs
• On the gamma-vector of symmetric edge polytopes
• $h^{\ast }$-vectors of graph polytopes using activities of dissecting spanning trees
• On the volume of sums of anti-blocking bodies
• Rigid Gorenstein toric Fano varieties arising from directed graphs