Modulus and characteristic of convexity: Difference between revisions

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==Definitions==

The '''modulus of convexity''' of a Banach space (''X'', ||&middotsdot;||) is the function {{nowrap|''δ'' : [0, 2] → [0, 1]}} defined by

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

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* When ''X'' is uniformly convex, it admits an equivalent norm with power type modulus of convexity.<ref>see {{citation

| last=Pisier |first=Gilles |author-link=Gilles Pisier

| title= Martingales with values in uniformly convex spaces | journal=[[Israel Journal of Mathematics]] | volume=20 | year=1975 | issue=3–4 | pages=326–350 | doi = 10.1007/BF02760337 | doi-access=free | mr=394135|s2cid=120947324 }}

.</ref> Namely, there exists {{nowrap|''q'' &ge; 2}} and a constant&nbsp;{{nowrap|''c'' &gt; 0}} such that

::<math>\delta(\varepsilon) \ge c \, \varepsilon^q, \quad \varepsilon \in [0, 2].</math>

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:<math>\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>\delta_p(\varepsilon+)=0,</math> one can suppose that <math>\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>\varepsilon=0</math>, one can calculate the <math>a_i</math> coefficients:

:<math>\delta_p(\varepsilon)=\frac{p-1}{8}\varepsilon^2+\frac{1}{384}(3-10p+9p^2-2p^3)\varepsilon^4+\cdots.

</math>