Compactness of composition operators on the Bergman spaces of convex domains and analytic discs

We study the compactness of composition operators on the Bergman spaces of certain bounded convex domains in ${\mathbb{C}}^n$ with non-trivial analytic discs contained in the boundary. As a consequence we characterize that compactness of the composition operator with a continuous symbol (up to the closure) on the Bergman space of the polydisc.