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 Template:Nowrap defined by

$\delta (\epsilon )=\inf \{1-\Vert \frac{x+y}{2}\Vert :x,y\in S,\Vert x-y\Vert \ge \epsilon \},$

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 Template:Nowrap and Template:Nowrap.

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

${\epsilon }_0=\sup \{\epsilon :\delta (\epsilon )=0\}.$

These notions are implicit in the general study of uniform convexity by J. A. Clarkson (Template:Harvtxt; 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 Template:Nowrap is also non-decreasing on Template:Nowrap. 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

$\delta (\epsilon /2)\le {\delta }_1(\epsilon )\le \delta (\epsilon ),\epsilon \in [0,2].$

• The normed space Template:Nowrap is uniformly convex if and only if its characteristic of convexity ε₀ is equal to 0, i.e., if and only if Template:Nowrap for every Template:Nowrap.
• The Banach space Template:Nowrap 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 Template:Nowrap and a constant Template:Nowrap such that

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

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:

${(1-{\delta }_p(\epsilon )+\frac{\epsilon }{2})}^p+{(1-{\delta }_p(\epsilon )-\frac{\epsilon }{2})}^p=2.$

Knowing that ${\delta }_p(\epsilon +)=0,$ one can suppose that ${\delta }_p(\epsilon )=a_0\epsilon +a_1{\epsilon }^2+\cdots$. Substituting this into the above, and expanding the left-hand-side as a Taylor series around $\epsilon =0$, one can calculate the $a_i$ coefficients:

${\delta }_p(\epsilon )=\frac{p-1}{8}{\epsilon }^2+\frac{1}{384}(3-10p+9p^2-2p^3){\epsilon }^4+\cdots .$

For $2<p<\mathrm{\infty }$, one has the explicit expression

${\delta }_p(\epsilon )=1-{(1-{\left(\frac{\epsilon }{2}\right)}^p)}^{\frac{1}{p}}.$

Therefore, ${\delta }_p(\epsilon )=\frac{1}{p{2}^p}{\epsilon }^p+\cdots$.

See also

• Uniformly smooth space

References

• Template:Cite book
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• 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. Template:MR
• Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.
• Template:Citation.
• 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.