, 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
definition of uniform convexity
as the modulus of continuity
does to the ε
definition of continuity
The modulus of convexity of a Banach space (X, || ||) is the function δ : [0, 2] → [0, 1] defined by
where B denotes the closed unit ball of (X, || ||). The characteristic of convexity of the space (X, || ||) is the number ε0 defined by
- 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.
- (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.
- Goebel, Kazimierz (1970). "Convexity of balls and fixed-point theorems for mappings with nonexpansive square". Compositio Mathematica 22 (3): 269–274.