# Integer polyhedron

From Egres Open

A polyhedron in [math]{\mathbb Z}^d[/math] 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 [math]u,v[/math] of an integer polytope, the associated
**primitive edge vector**is [math]\frac{u-v}{\mathrm{gcd}(u-v)}[/math]. - A full-dimensional polytope is
**simple**if every vertex has*d*neighbours. - A simple integer polytope is
**smooth**if for every vertex*v*the primitive edge vectors from*v*form a basis of the lattice [math]{\mathbb Z}^d[/math].

A summary of open questions in this topic can be found on the page Integer polyhedra.