On Integer Points in Polyhedra: A Lower Bound

Given a polyhedron P subset Rn we write PI for the convex hull of the integral points in P. It is known that PI can have at most O(ϕn-1) vertices if P is a rational polyhedron with size ϕ. Here we give an example showing that PI can have as many as Ω(ϕn-1) vertices. The construction uses the Dirichlet unit theorem.