  In this paper we study the automorphism group of smoothly bounded convex
domains. We show that such a domain is biholomorphic to a "polynomial
ellipsoid" (that is, a domain defined by a weighted homogeneous balanced
polynomial) if and only if the limit set of the automorphism group intersects
at least two closed complex faces of the set. The proof relies on a detailed
study of the geometry of the Kobayashi metric and ideas from the theory of
non-positively curved metric spaces. We also obtain a number of other results
including the Greene-Krantz conjecture in the case of uniform non-tangential
convergence, new results about continuous extensions (of biholomorphisms and
complex geodesics), and a new Wolff-Denjoy theorem.
