The minimal volume of a lattice polytope

Abstract

Let $\mathcal{P}\subset {\mathbb{R}}^d$ be a lattice polytope of dimension $d$. Let $b$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c$ that to the interior of $\mathcal{P}$. It follows from a lower bound theorem of Ehrhart polynomials that, when $c>0$, the volume of $\mathcal{P}$ is bigger than or equal to $(dc+(d-1)b-d^2+2)/d!$. In the present paper, via triangulations, a short and elementary proof of the minimal volume formula is given.