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

中文翻译：

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局部反阻塞格多胞体的 $$h^*$$-多项式及其 $$\gamma $$-Positivity

格多面体 $$\mathscr {P} \subset \mathbb {R}^d$$ 被称为局部反阻塞多面体，如果对于任何封闭的 orthant $${\mathbb R}^d_{\varepsilon }$$ $$\mathbb {R}^d$$ , $$\mathscr {P} \cap \mathbb {R}^d_{\varepsilon }$$ 单模等价于坐标超平面反射的反阻塞多胞体。我们给出了局部反阻塞晶格多胞体的 $$h^*$$ -多项式的公式。特别地，我们讨论了局部反阻塞自反多胞体的 $$h^*$$ 多项式的 $$\gamma $$ -正性。