The $$h^*$$-Polynomials of Locally Anti-Blocking Lattice Polytopes and Their $$\gamma $$-Positivity

Abstract

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