Uniform Convexity and Uniform Smoothness

We begin this chapter by describing the concept of uniform convexity of a Banach space.1

Definition 3.1.1. Let X be a Banach space of dimension ≥ 2. The modulus of convexity of X is defined by the formula:

X is said to be uniformly convex if δX (ǫ) > 0, for ǫ > 0. X is said to be q-uniformly convex if there exists a constant C such that δX (ǫ) ≥ Cǫq, q ≥ 2.