Modulus and characteristic of convexity explained

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

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

where S denotes the unit sphere of (X, || ||). In the definition of δ(ε), one can as well take the infimum over all vectors x, y in X such that and .^[1]

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 (; 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 .^[2] The modulus of convexity need not itself be a convex function of ε.^[3] However, the modulus of convexity is equivalent to a convex function in the following sense:^[4] there exists a convex function δ₁(ε) such that

\delta(\varepsilon / 2 ) \le \delta _{1(\varepsilon)} \le \delta(\varepsilon), \varepsilon \in [ 0 , 2 ].

• 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.^[5] Namely, there exists and a constant  such that

\delta(\varepsilon) \ge c \varepsilon ^{q,} \varepsilon \in [ 0 , 2 ].

Modulus of convexity of the L^P spaces

The modulus of convexity is known for the L^P spaces. If

1 <p\le 2 , then it satisfies the following implicit equation:\left( 1- \delta _{p(\varepsilon) + (\varepsilon)/(2)} \right) ^{p + \left( 1- \delta} _{p(\varepsilon) - (\varepsilon)/(2)} \right) ^{p=2.} Knowing that \delta _{p(\varepsilon+)=0,} one can suppose that \delta _{p(\varepsilon)=a} _{0\varepsilon+a} ^{2+ …} ^{1 \varepsilon} . Substituting this into the above, and expanding the left-hand-side as a Taylor series around \varepsilon =0 , one can calculate the a _{i} coefficients:\delta _{p(\varepsilon) = (p -1)/(8)} \varepsilon ^{2+ (1)/(384)} ( 3-10 p +9 p ^{2-2p} ^{3)\varepsilon} ^{4+ … .}

For

2 <p< inf ty , one has the explicit expression\delta _{p(\varepsilon) =1- \left( 1- \left( (\varepsilon)/(2)} \right) ^{p\right)} ^{(1 p)/()} . Therefore, \delta _{p(\varepsilon) = (1)/(p 2 ^{p})} \varepsilon ^{p+ …} .

See also

• Uniformly smooth space

References

• Book: Beauzamy, Bernard. Introduction to Banach Spaces and their Geometry. 1985 . 1982. Second revised. North-Holland. 889253. 0-444-86416-4.
  • 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.

Notes and References

1. p. 60 in .
2. Lemma 1.e.8, p. 66 in .
3. see Remarks, p. 67 in .
4. see Proposition 1.e.6, p. 65 and Lemma 1.e.7, 1.e.8, p. 66 in .
5. see .