Uniformly convex functions on Banach spaces

Abstract

Given a Banach space (X,∥·∥), we study the connection between uniformly convex functions f : X → ℝ bounded above by ∥·∥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 an equivalent norm with modulus of convexity of power type 2. © 2008 American Mathematical Society.