A note on palindromic $\delta$-vectors for certain rational polytopes

TL;DR

This paper proves that the Ehrhart δ-vector of certain rational polytopes containing the origin and with a lattice dual is palindromic, extending Hibi's theorem from lattice to rational polytopes using an elementary proof.

Contribution

It provides the first elementary lattice-point proof that rational polytopes with lattice duals have palindromic Ehrhart δ-vectors, extending known results.

Findings

Ehrhart δ-vector of rational polytopes with lattice duals is palindromic

Elementary proof method for palindromicity

Extension of Hibi's theorem to rational polytopes

Abstract

Let P be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi's Palindromic Theorem tells us that if P is also a lattice polytope then the Ehrhart $\delta$-vector of P is palindromic. Perhaps less well-known is that a similar result holds when P is rational. We present an elementary lattice-point proof of this fact.