Modulus and characteristic of convexity

In mathematics, the modulus of convexity and the 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 defined by

where S denotes the unit sphere of (X, || ||). In the definition of ÃŽÂ´(ε), one can as well take the infimum over all vectors x, y in X such that and .

The characteristic of convexity of the space (X, || ||) is the number ε₀ defined by

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

Properties

• The modulus of convexity, δ(ε), is a non-decreasing function of ε, and the quotient is also non-decreasing on . The modulus of convexity need not itself be a convex function of ÃŽÂµ. However, the modulus of convexity is equivalent to a convex function in the following sense: there exists a convex function δ₁(ε) such that

:

• The normed space is uniformly convex if and only if its characteristic of convexity ε₀ is equal to 0, i.e., if and only if for every .
• The Banach space 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, i.e., if only antipodal points (of the form x and y = −x) of the unit sphere can have distance equal to 2.
• When X is uniformly convex, it admits an equivalent norm with power type modulus of convexity. Namely, there exists and a constant  such that

:

Modulus of convexity of the L^P spaces

The modulus of convexity is known for the L^P spaces. If , then it satisfies the following implicit equation:

Knowing that one can suppose that . Substituting this into the above, and expanding the left-hand-side as a Taylor series around , one can calculate the coefficients:

For , one has the explicit expression

Therefore, .

See also

• Uniformly smooth space

Notes

References

• Fuster, Enrique Llorens. Some moduli and constants related to metric fixed point theory. Handbook of metric fixed point theory, 133–175, Kluwer Acad. Publ., Dordrecht, 2001.
• 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.