m$. Given a positive integer $t$, let $t{\mathcal{P}}\coloneqq \{t\x \mid \x \in {\mathcal{P}}\}$ be the $t$-th dilate of ${\mathcal{P}}$. The lattice point enumeration function $$\operatorname{ehr}_{\mathcal{P}}(t)\coloneqq \#(t{\mathcal{P}}\cap {\mathbb{Z}}^m)$$ is called the [**Ehrhart polynomial**]{} of ${\mathcal{P}}$. By a classical result of Ehrhart [@Ehrhart], this polynomial agrees with a polynomial of degree $d$ in the variable $t$. Equivalently, one may also consider the [**Ehrhart series**]{} of ${\mathcal{P}}$ which is defined to be the formal power series $$\operatorname{Ehr}_{\mathcal{P}}(z)\coloneqq 1+ \sum_{t\geq 1}\operatorname{ehr}_{\mathcal{P}}(t)z^t=\frac{1+h_1^\ast z+\cdots +h_{d-1}^\ast z^{d-1}+h_d^\ast z^d}{(1-z)^{d+1}}.$$ The numerator of the Ehrhart series is called the [**$h^\ast$-polynomial**]{} and the vector of coefficients $h^\ast({\mathcal{P}})=(1,h^\ast_1,\ldots, h^\ast_d)$ the [**$h^\ast$-vector**]{}. By a result of Stanley [@Stanley-Hilbert-Functions-CM], $(1,h^\ast_1,\ldots, h^\ast_d)\in {\mathbb{Z}}_{\geq 0}^{d+1}$.

Studying $h^\ast({\mathcal{P}})$ often informs the algebraic and geometric structure of a lattice polytope ${\mathcal{P}}$. If $\mathbf{0}$ is in the interior of ${\mathcal{P}}$, the [**(polar) dual polytope**]{} of ${\mathcal{P}}$ is the rational polytope $${\mathcal{P}}^\ast\coloneqq \left\{\y\in{\mathbb{R}}^m \mid \langle\y,\x\rangle\leq 1 \mbox{ for all } \x\in {\mathcal{P}}\right\}.$$ A polytope ${\mathcal{P}}$ with $\mathbf{0}$ in its interior is called [**reflexive**]{} if ${\mathcal{P}}^\ast$ is a lattice polytope. Equivalently, ${\mathcal{P}}$ is reflexive (up to translation and lattice preserving transformations) if it contains a unique lattice point $\p$ in its interior, and for any facet ${\mathcal{P}}\cap\{\x\in{\mathbb{R}}^m \mid \langle\a,\x\rangle= b\}$ 