One of the most important problems in the field of several complex variables is the Greene-Krantz conjecture:

Conjecture Let D be a smoothly bounded domain in C^{n with non-compact automorphism group. Then the boundary of D is of finite type at any boundary orbit accumulation point.}

The purpose of this dissertation is to prove a result that supports the truthfulness of this conjecture:

Theorem Let D be a smoothly bounded convex domain in C^{n. Suppose there exists a point p in D and a sequence of automorphisms of D, f _{j, such that f j(p) &rarr q in the boundary of D non-tangentially. Furthermore, suppose Condition LTW holds. Then, the boundary of D is variety-free at q.}}