Integer polyhedron

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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.