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In [[mathematics]], the '''modulus of convexity''' and the '''characteristic of convexity''' are measures of "how [[convex set|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 [[uniformly convex space|uniform convexity]] as the [[modulus of continuity]] does to the ''ε''-''δ'' definition of [[continuous function|continuity]].
==Definitions==
The '''modulus of convexity''' of a Banach space (''X'', ||&
:<math>\delta (\varepsilon) = \inf \left\{
where ''S'' denotes the unit sphere of (''X'', || ||).
The '''characteristic of convexity''' of the space (''X'', || ||) is the number ''ε''<sub>0</sub> defined by
:<math>\varepsilon_{0} = \sup \{ \varepsilon \,:\, \delta(\varepsilon) = 0 \}.</math>
These notions are implicit in the general study of uniform convexity by J. A. Clarkson ({{harvtxt|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.<ref>{{citation
| last = Day
| first = Mahlon
| title = Uniform convexity in factor and conjugate spaces
| journal = Annals of Mathematics |series = 2
| volume = 45
| year = 1944
| pages = 375–385
| doi = 10.2307/1969275
| issue = 2
| jstor = 1969275
}}</ref>
==Properties==
* The modulus of convexity, ''δ''(''ε''), is a [[monotonic function|non-decreasing]] function of ''ε'', and the quotient {{nowrap|''δ''(''ε'') / ''ε''}} is also non-decreasing on {{nowrap|(0, 2]}}.<ref>Lemma 1.e.8, p. 66 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref> The modulus of convexity need not itself be a [[convex function]] of ''ε''.<ref>see Remarks, p. 67 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref> However, the modulus of convexity is equivalent to a convex function in the following sense:<ref>see Proposition 1.e.6, p. 65 and Lemma 1.e.7, 1.e.8, p. 66 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref> there exists a convex function ''δ''<sub>1</sub>(''ε'') such that
::<math>\delta(\varepsilon / 2) \le \delta_1(\varepsilon) \le \delta(\varepsilon), \quad \varepsilon \in [0, 2].</math>
* The normed space {{nowrap|(''X'', ǁ ⋅ ǁ)}} is [[uniformly convex space|uniformly convex]] [[if and only if]] its characteristic of convexity ''ε''<sub>0</sub> is equal to 0, ''i.e.'', if and only if {{nowrap|''δ''(''ε'') > 0}} for every {{nowrap|''ε'' > 0}}.
* The Banach space {{nowrap|(''X'', ǁ&
* 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= | mr=394135|s2cid=120947324 }}
.</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 ==
*[[Uniformly smooth space]]
==Notes==
{{reflist}}
==References==
* {{cite book|author=Beauzamy, Bernard|title=Introduction to Banach Spaces and their Geometry|year=1985 |orig-year=1982|edition=Second revised|publisher=North-Holland|mr=889253|isbn=0-444-86416-4}}
*{{citation
| last = Clarkson
| first = James
| title = Uniformly convex spaces
| journal =
| volume = 40
| year = 1936
| pages = 396–414
| doi = 10.2307/1989630
| issue = 3
| 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'', 133–175, Kluwer Acad. Publ., Dordrecht, 2001. {{MR|1904276}}
* [[Joram Lindenstrauss|Lindenstrauss, Joram]] and Benyamini, Yoav. ''Geometric nonlinear functional analysis'' Colloquium publications, 48. American Mathematical Society.
*{{citation
| last1 = Lindenstrauss
| first1 = Joram | author1-link = Joram Lindenstrauss
| last2 = Tzafriri | first2 = Lior
| title = Classical Banach spaces. II. Function spaces
| series = Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]
| volume = 97
| publisher = Springer-Verlag
| ___location = Berlin-New York
| year = 1979
| pages = x+243
| 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, 73–149, 1971; ''Russian Math. Surveys'', v. 26 6, 80–159.
{{Banach spaces}}
{{Functional analysis}}
[[Category:Banach spaces]]
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