  In this paper we prove: if the complete K\"ahler-Einstein metric on a bounded
convex domain (with no boundary regularity assumptions) is Gromov hyperbolic,
then the $\bar{\partial}$-Neumann problem satisfies a subelliptic estimate.
This is accomplished by constructing bounded plurisubharmonic function whose
Hessian grows at a certain rate (which implies a subelliptic estimate by work
of Catlin and Straube). We also provide a characterization of Gromov
hyperbolicity in terms of orbit of the domain under the group of affine
transformations. This characterization allows us to construct many examples.
For instance, if the Hilbert metric on a bounded convex domain is Gromov
hyperbolic, then the K\"ahler-Einstein metric is as well.
