The H-representation of a convex polytope $S$, is just a set of linear inequalities corresponding to the intersection of halfspaces: $S = ( x | Ax\leq b )$.
One could also represent a convex polytope as the convex-hull of its vertices, called the $V$-representation: $S = \mathrm{conv}(a_1,..., a_m)$.
It is known that sometimes the H-representation is much more efficient than the V-representation: for example, the unit cube, with $2n$ faces and $2^n$ vertices, whereas in other cases, such as the crosspolytope the V-representation of $2n$ vertices is much more efficient than the H-representation, which is exponential in $n$.
My question is, is there any general characterization of V-polytopes whose H-description is exponential, or vice versa?