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

:$\delta (\varepsilon) = \inf \left\,$

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 .

The characteristic of convexity of the space (''X'', , ,  , , ) is the number ''ε''₀ defined by

:$\varepsilon_ = \sup \.$

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 ''ε''. However, the modulus of convexity is equivalent to a convex function in the following sense: there exists a convex function ''δ''₁(''ε'') such that
::$\delta(\varepsilon / 2) \le \delta_1(\varepsilon) \le \delta(\varepsilon), \quad \varepsilon \in$, 2

* The normed space is uniformly convex if and only if its characteristic of convexity ''ε''₀ 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
::$\delta(\varepsilon) \ge c \, \varepsilon^q, \quad \varepsilon \in$, 2

Modulus of convexity of the ''L''^''P'' spaces

The modulus of convexity is known for the ''L''^''P'' spaces. If 1, then it satisfies the following implicit equation:

:$\left(1-\delta_p(\varepsilon)+\frac\right)^p+\left(1-\delta_p(\varepsilon)-\frac\right)^p=2.$
Knowing that $\delta_p(\varepsilon+)=0,$ one can suppose that $\delta_p(\varepsilon)=a_0\varepsilon+a_1\varepsilon^2+\cdots$. 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:
:$\delta_p(\varepsilon)=\frac\varepsilon^2+\frac(3-10p+9p^2-2p^3)\varepsilon^4+\cdots.$

For 2, one has the explicit expression
:$\delta_p(\varepsilon)=1-\left(1-\left(\frac\right)^p\right)^.$
Therefore, $\delta_p(\varepsilon)=\frac\varepsilon^p+\cdots$.

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.

{{Functional analysis

Banach spaces
Convex analysis