Modulus and characteristic of convexity

In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε-δ definition of uniform convexity as the modulus of continuity does to the ε-δ definition of continuity. The modulus of convexity of a Banach space (X, ||·||) is the function δ : [0, 2] → [0, 1] defined by where S denotes the unit sphere of (X, || ||). In the definition of δ(ε), one can as well take the infimum over all vectors x, y in X such that ǁxǁ, ǁyǁ ≤ 1 and ǁx − yǁ ≥ ε. The characteristic of convexity of the space (X, || ||) is the number ε0 defined by These notions are implicit in the general study of uniform convexity by J. A. Clarkson (; this is the same paper containing the statements of Clarkson's inequalities). The term "modulus of convexity" appears to be due to M. M. Day. The modulus of convexity, δ(ε), is a non-decreasing function of ε, and the quotient δ(ε) / ε is also non-decreasing on (0, 2]. The modulus of convexity need not itself be a convex function of ε. However, the modulus of convexity is equivalent to a convex function in the following sense: there exists a convex function δ1(ε) such that The normed space (X, ǁ ⋅ ǁ) is uniformly convex if and only if its characteristic of convexity ε0 is equal to 0, i.e., if and only if δ(ε) > 0 for every ε > 0. The Banach space (X, ǁ ⋅ ǁ) is a strictly convex space (i.e., the boundary of the unit ball B contains no line segments) if and only if δ(2) = 1, i.e., if only antipodal points (of the form x and y = −x) of the unit sphere can have distance equal to 2. When X is uniformly convex, it admits an equivalent norm with power type modulus of convexity. Namely, there exists q ≥ 2 and a constant c > 0 such that The modulus of convexity is known for the LP spaces. If , then it satisfies the following implicit equation: Knowing that one can suppose that .