Reverse Chvátal-Gomory Rank

We introduce the reverse Chvatal--Gomory rank $r^*(P)$ of an integral polyhedron $P$, defined as the supremum of the Chvatal--Gomory ranks of all rational polyhedra whose integer hull is $P$. A well-known example in dimension two shows that there exist integral polytopes $P$ with $r^*(P)=+\infty$. We provide a geometric characterization of polyhedra with this property in every dimension, and investigate upper bounds on $r^*(P)$ when this value is finite.