# The $h^*$-polynomials of locally anti-blocking lattice polytopes and their $\gamma$-positivity

## TL;DR

This paper investigates the $h^*$-polynomials of locally anti-blocking lattice polytopes, providing formulas and exploring their $ heta$-positivity, especially in the context of reflexive polytopes, advancing understanding of their combinatorial properties.

## Contribution

It introduces a formula for the $h^*$-polynomials of locally anti-blocking lattice polytopes and examines their $ heta$-positivity, particularly for reflexive cases, revealing new structural insights.

## Key findings

- Derived a formula for $h^*$-polynomials of locally anti-blocking polytopes
- Established $ heta$-positivity for $h^*$-polynomials of reflexive polytopes
- Enhanced understanding of combinatorial properties of these polytopes

## Abstract

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