Modulus and characteristic of convexity

In mathematics, the modulus and 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.

[edit] Definitions

The modulus of convexity of a Banach space (X, || ||) is the function δ : [0, 2] → [0, 1] defined by

$\delta (\varepsilon) = \inf \left\{ \left. 1 - \left\| \frac{x + y}{2} \right\| \, \right| x, y \in B, \| x - y \| \geq \varepsilon \right\},$

where B denotes the closed unit ball of (X, || ||). The characteristic of convexity of the space (X, || ||) is the number ε₀ defined by

$\varepsilon_{0} = \sup \{ \varepsilon | \delta(\varepsilon) = 0 \}.$

The modulus of convexity, δ(ε), is a non-decreasing function of ε. Goebel claims the modulus of convexity is itself convex, while Lindenstrauss and Tzafriri claim that the modulus of convexity need not itself be a convex function of ε.^[1]
(X, || ||) is a uniformly convex space if and only if its characteristic of convexity ε₀ = 0.
(X, || ||) 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.

^ p. 67 in Lindenstrauss, Joram; Tzafriri, Lior Classical Banach spaces. II. Function spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 97. Springer-Verlag, Berlin-New York, 1979. x+243 pp.

Goebel, Kazimierz (1970). "Convexity of balls and fixed-point theorems for mappings with nonexpansive square". Compositio Mathematica 22 (3): 269–274.