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: $S = conv(a_1,..., a_m)$.

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?