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==Definitions==
The '''modulus of convexity''' of a Banach space (''X'', ||&
:<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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| first = Mahlon
| title = Uniform convexity in factor and conjugate spaces
| journal =
| volume = 45
| year = 1944
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| doi = 10.2307/1969275
| issue = 2
| jstor = 1969275
}}</ref>
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* The Banach space {{nowrap|(''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 point]]s (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.<ref>see {{citation
| last=Pisier |first=Gilles |
| title= Martingales with values in uniformly convex spaces | journal=[[Israel
.</ref> Namely, there exists {{nowrap|''q'' ≥ 2}} and a constant {{nowrap|''c'' > 0}} such that
::<math>\delta(\varepsilon) \ge c \, \varepsilon^q, \quad \varepsilon \in [0, 2].</math>
==Modulus of convexity of the ''L''<sup>''P''</sup> spaces==
The modulus of convexity is known for the ''L''<sup>''P''</sup> spaces.<ref>{{citation
| last = Hanner
| first = Olof
| title = On the uniform convexity of <math>L^p</math> and <math>\ell^p</math>
| journal = Arkiv för Matematik
| volume = 3
| year = 1955
| pages = 239–244
| doi = 10.1007/BF02589410
| doi-access = free
}}</ref> If <math>1<p\le2</math>, then it satisfies the following implicit equation:
:<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>
For <math>2<p<\infty</math>, one has the explicit expression
:<math>\delta_p(\varepsilon)=1-\left(1-\left(\frac{\varepsilon}{2}\right)^p\right)^{\frac1p}.
</math>
Therefore, <math>\delta_p(\varepsilon)=\frac{1}{p2^p}\varepsilon^p+\cdots</math>.
== See also ==
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==References==
* {{cite book|author=Beauzamy, Bernard|title=Introduction to Banach Spaces and their Geometry|year=1985 |
*{{citation
| last = Clarkson
| first = James
| title = Uniformly convex spaces
| journal =
| volume = 40
| year = 1936
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| publisher = American Mathematical Society
| jstor = 1989630
| doi-access = free
}}
* Fuster, Enrique Llorens. Some moduli and constants related to metric fixed point theory. ''Handbook of metric fixed point theory'',
* [[Joram Lindenstrauss|Lindenstrauss, Joram]] and Benyamini, Yoav. ''Geometric nonlinear functional analysis'' Colloquium publications, 48. American Mathematical Society.
*{{citation
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| isbn = 3-540-08888-1
}}.
* [[Vitali Milman|Vitali D. Milman]]. Geometric theory of Banach spaces II. Geometry of the unit sphere. ''Uspechi Mat. Nauk,'' vol. 26, no. 6,
{{Banach spaces}}
{{Functional analysis}}
[[Category:Banach spaces]]
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