In the standard word-RAM and cell-probe models of computation, we prove that the optimal query time for dynamic convex hulls is, in fact, Θ(lg n / lg lg n), for polylogarithmic update time (and word size). Our lower bound is based on a reduction from the marked-ancestor problem, and is one of the first data structural lower bounds for a nonorthogonal geometric problem. Our upper bounds follow a recent trend of attacking nonorthogonal geometric problems from an information-theoretic perspective that has proved central to advanced data structures. Interestingly, our upper bounds are the first to successfully apply this perspective to dynamic geometric data structures, and require substantially different ideas from previous work.