where $\a$ is primitive (meaning the greatest common divisor of the coordinates of $\a$ is 1) we have $|\langle\a,\p\rangle -b\,|=1$. This final quantity is called the [**lattice distance**]{} of $\p$ from the facet.

Let $\mathcal{P}$ be a lattice polytope of dimension $d$ containing the origin in its interior and having Ehrhart series $$\textnormal{Ehr}_{\mathcal{P}}(z) = \frac{h_0^{\ast}+h_1^\ast z+\cdots +h_{d-1}^\ast z^{d-1}+h_d^\ast z^d}{(1-z)^{d+1}}.$$ Then $\mathcal{P}$ is reflexive if and only if $h_i^{\ast}=h_{d-i}^{\ast}$ for all $0\leq i \leq \lfloor\frac{d}{2}\rfloor$.

In other words, the $h^\ast$-polynomial of a reflexive lattice polytope is a palindromic polynomial of degree $d$.

A relaxation of reflexivity is the Gorenstein property. We say that ${\mathcal{P}}$ is [**Gorenstein**]{} if there is some positive integer $c$ such that $c{\mathcal{P}}$ is a reflexive polytope, and the integer $c$ is called the [**Gorenstein index**]{} of ${\mathcal{P}}$. Similarly, this is completely detected by the Ehrhart series as ${\mathcal{P}}$ is Gorenstein if and only if its $h^\ast$-polynomial is palindromic of degree $d-c+1$ by a result of De Negri and Hibi [@DeNegri-Hibi].

Given a lattice polytope ${\mathcal{P}}$, one can consider the interplay between the convex geometry of ${\mathcal{P}}$ in ${\mathbb{R}}^m$ with the induced arithmetic structure of ${\mathcal{P}}\cap {\mathbb{Z}}^m$. This motivates the discussion of triangulations and the integer decomposition property. 