# Integer polyhedron

A polyhedron in ${\mathbb Z}^d$ is integer (or integral) if every minimal face contains an integer point. In particular, a polytope is integral if and only if all of its vertices are integral. Integer polytopes are also sometimes called lattice polytopes. Some related definitions:
• For two neighbouring vertices $u,v$ of an integer polytope, the associated primitive edge vector is $\frac{u-v}{\mathrm{gcd}(u-v)}$.
• A simple integer polytope is smooth if for every vertex v the primitive edge vectors from v form a basis of the lattice ${\mathbb Z}^d$.