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

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

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

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

These notions are implicit in the general study of uniform convexity by J. A. Clarkson (see below; 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 (see reference below).

Properties

• The modulus of convexity, δ(ε), is a non-decreasing function of ε. (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

• Beauzamy, Bernard (1985 [1982]). Introduction to Banach Spaces and their Geometry (Second revised ed.). North-Holland. {{cite book}}: Check date values in: |year= (help)CS1 maint: year (link)

• Clarkson, James (1936). "Uniformly convex spaces". Trans. Amer. Math. Soc. 40 (3). American Mathematical Society: 396–414. doi:10.2307/1989630.

• Day, Mahlon (1944). "Uniform convexity in factor and conjugate spaces". Ann. Of Math. (2). 45 (2). Annals of Mathematics: 375–385. doi:10.2307/1969275.
• Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.
• Vitali D. Milman. Geometric theory of Banach spaces II. Geometry of the unit sphere. Uspechi Mat. Nauk, vol. 26, no. 6, 73-149, 1971; Russian Math. Surveys, v. 26 6, 80-159.