Definitions

# 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.

## Definitions

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

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

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

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

## Properties

• 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.

## References

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