  We give a necessary complex geometric condition for a bounded smooth convex
domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic.
More precisely, we prove that if a smooth bounded convex domain contains an
analytic disk in its boundary, then the domain is not Gromov hyperbolic for the
Kobayashi distance. We also provide examples of bounded smooth convex domains
that are not strongly pseudoconvex but are Gromov hyperbolic.
