UNIFORMLY CONVEX FUNCTIONS ON BANACH SPACES

We study the connection between uniformly convex functions f : X -> R bounded above by ||x||^p, and the existence of norms on X with moduli of convexity of power type. In particular, we show that there exists a uniformly convex function f : X -> R bounded above by ||.||2 if and only if X admits a norm with modulus of convexity of power type 2.