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 δ : [0, 2] → [0, 1] defined by

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

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 ǁxǁ, ǁyǁ ≤ 1 and ǁx − yǁ ≥ ε.

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

[math]\displaystyle{ \varepsilon_{0} = \sup \{ \varepsilon \,:\, \delta(\varepsilon) = 0 \}. }[/math]

These notions are implicit in the general study of uniform convexity by J. A. Clarkson ((Clarkson 1936); 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 δ(ε) / ε is also non-decreasing on (0, 2]. The modulus of convexity need not itself be a convex function of ε. However, the modulus of convexity is equivalent to a convex function in the following sense: there exists a convex function δ₁(ε) such that

[math]\displaystyle{ \delta(\varepsilon / 2) \le \delta_1(\varepsilon) \le \delta(\varepsilon), \quad \varepsilon \in [0, 2]. }[/math]

• The normed space (X, ǁ ⋅ ǁ) is uniformly convex if and only if its characteristic of convexity ε₀ is equal to 0, i.e., if and only if δ(ε) > 0 for every ε > 0.
• The Banach space (X, ǁ ⋅ ǁ) 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 q ≥ 2 and a constant c > 0 such that

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

Modulus of convexity of the L^P spaces

The modulus of convexity is known for the L^P spaces. If [math]\displaystyle{ 1\lt p\le2 }[/math], then it satisfies the following implicit equation:

[math]\displaystyle{ \left(1-\delta_p(\varepsilon)+\frac{\varepsilon}{2}\right)^p+\left(1-\delta_p(\varepsilon)-\frac{\varepsilon}{2}\right)^p=2. }[/math]

Knowing that [math]\displaystyle{ \delta_p(\varepsilon+)=0, }[/math] one can suppose that [math]\displaystyle{ \delta_p(\varepsilon)=a_0\varepsilon+a_1\varepsilon^2+\cdots }[/math]. Substituting this into the above, and expanding the left-hand-side as a Taylor series around [math]\displaystyle{ \varepsilon=0 }[/math], one can calculate the [math]\displaystyle{ a_i }[/math] coefficients:

[math]\displaystyle{ \delta_p(\varepsilon)=\frac{p-1}{8}\varepsilon^2+\frac{1}{384}(3-10p+9p^2-2p^3)\varepsilon^4+\cdots. }[/math]

For [math]\displaystyle{ 2\lt p\lt \infty }[/math], one has the explicit expression

[math]\displaystyle{ \delta_p(\varepsilon)=1-\left(1-\left(\frac{\varepsilon}{2}\right)^p\right)^{\frac1p}. }[/math]

Therefore, [math]\displaystyle{ \delta_p(\varepsilon)=\frac{1}{p2^p}\varepsilon^p+\cdots }[/math].

See also

• Uniformly smooth space

References

• Beauzamy, Bernard (1985). Introduction to Banach Spaces and their Geometry (Second revised ed.). North-Holland. ISBN 0-444-86416-4. 
• Clarkson, James (1936), "Uniformly convex spaces", Transactions of the American Mathematical Society (American Mathematical Society) 40 (3): 396–414, doi:10.2307/1989630 
• 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. MR1904276
• Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.
• Lindenstrauss; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 3-540-08888-1 .
• 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.