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

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

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

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

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.^[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.

References

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.
• Beauzamy, Bernard (1985 [1982]). Introduction to Banach Spaces and their Geometry (Second revised ed.). North-Holland. {{cite book}}: Check date values in: |year= (help)
• Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.