On integer points in polyhedra: A lower bound (Q1196682)

For a polyhedron \(P\subset\mathbb{R}^ n\) let \(P_ I\) denote the convex hull of integer points in \(P\). It is known that \(P_ I\) can have at most \(O(\varphi^{n-1})\) vertices if \(P\) is a rational polyhedron with size \(\varphi\). Here the size of a rational polyhedron is the sum of the sizes of the defining inequalities, measured by the encoding lengths as binary strings. The authors give an example which shows that \(P_ I\) can have as many as \(\Omega(\varphi^{n-1})\) vertices.