Lattice polytopes with the minimal volume

Let $\mathcal{P}\subset {\mathbb{R}}^d$ be a lattice polytope of dimension $d$. Let $b(\mathcal{P})$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c(\mathcal{P})$ that to the interior of $\mathcal{P}$. It follows from the lower bound theorem of Ehrhart polynomials that, when $c>0$, $$\mathrm{v}\mathrm{o}\mathrm{l}(\mathcal{P})\ge (d\cdot c(\mathcal{P})+(d-1)\cdot b(\mathcal{P})-d^2+2)\mathrm{/}d!,$$ where $\mathrm{v}\mathrm{o}\mathrm{l}(\mathcal{P})$ is the (Lebesgue) volume of $\mathcal{P}$. Pick’s formula guarantees that, when $d=2$, the above inequality is an equality. In the present paper several classes of lattice polytopes for which the equality here holds will be presented.