It is known that any rational convex polytope expressed as $\{ x\in\mathbb{R}^d : Ax \ge b \}$, where $A\in\mathbb{Z}^{k\times d}$ and $b\in\mathbb{Z}^k$, can be written as the convex hull of finitely many points. 


My question is, given the above representation in terms of hyperplanes, one can determine (easily) whether the polytope is integral ---that is, whether the polytope can be written as the convex hull of points in the integer lattice. 



Is there a simple test to determine whether a polytope is integral? 
An incomplete solution:


There is a polynomial-time test for total unimodularity, and if $A$ is totally unimodular, then the polytope is integral.


