Modulus and characteristic of convexity: Difference between revisions

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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'', ||&nbspsdot;||) is the function {{nowrap|''δ''  :  [0,  2]  →  [0,  1]}} defined by

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

where ''S'' denotes the unit sphere of (''X'',&nbsp;||&nbsp;||). The In the definition of&nbsp;''δ'characteristic of convexity'(''ε''), ofone can as well take the spaceinfimum over all vectors (''Xx'', ''y'' in&nbsp;''X'' such that {{nowrap||ǁ''x''ǁ, ǁ''y''ǁ &nbsple; 1}} and {{nowrap||)ǁ''x'' is&minus; the''y''ǁ number&ge; ''ε''}}.<subref>0p.&nbsp;60 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</subref> defined by

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

:<math>\varepsilon_{0} = \sup \{ \varepsilon | \delta(\varepsilon) = 0 \}.</math>

:<math>\varepsilon_{0} = \sup \{ \varepsilon |\,:\, \delta(\varepsilon) = 0 \}.</math>

These notions are implicit in the general study of uniform convexity by J. A. Clarkson (see below; 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 (see reference below).

These notions are implicit in the general study of uniform convexity by J. &nbsp;A. &nbsp;Clarkson (see below{{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. &nbsp;M. &nbsp;Day (see reference below).<ref>{{citation

| last = Day

| first = Mahlon

| title = Uniform convexity in factor and conjugate spaces

| publisherjournal = Annals of Mathematics |series = 2

| volume = 45

| year = 1944

| pages = 375&ndash;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'')&thinsp;/&thinsp;''ε''}} is also non-decreasing on&nbsp;{{nowrap|(0, 2]}}.<ref>Lemma 1.e.8, p.&nbsp;66 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref> The modulus of convexity need not itself be a [[convex function]] of&nbsp;''ε''.<ref>see Remarks, p.&nbsp;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.&nbsp;65 and Lemma 1.e.7, 1.e.8, p.&nbsp;66 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref> there exists a convex function ''δ''₁(''ε'') such that

::<math>\delta(\varepsilon / 2) \le \delta_1(\varepsilon) \le \delta(\varepsilon), \quad \varepsilon \in [0, 2].</math>

* (''X'',&nbsp;||&nbsp;||) is a [[uniformly convex space]] [[if and only if]] its characteristic of convexity ''ε''₀ is equal to&nbsp;0.

* (''X'',&nbsp;||&nbsp;||) is a [[strictly convex space]] (i.e., the boundary of the unit ball ''B'' contains no line segments) if and only if ''δ''(2)&nbsp;=&nbsp;1.

* The normed space {{nowrap|(''X'', ǁ&nbspthinsp;||&nbspsdot;||&thinsp;ǁ)}} is a [[uniformly convex space|uniformly convex]] [[if and only if]] its characteristic of convexity ''ε''₀ is equal to&nbsp;0, ''i.e.'', if and only if {{nowrap|''δ''(''ε'') > 0}} for every&nbsp;{{nowrap|''ε'' > 0}}.

==References==

* The Banach space {{nowrap|(''X'', ǁ&nbspthinsp;||&nbspsdot;||&thinsp;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)&nbsp;=&nbsp;1, ''i.e.'', if only [[antipodal point]]s (of the form ''x'' and ''y''&nbsp;=&nbsp;&minus;''x'') of the unit sphere can have distance equal to&nbsp;2.

* 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

* {{cite journal | last=Pisier |first=Gilles |authorlink=Gilles Pisier | title= Martingales with values in uniformly convex spaces | journal=[[Israel J.Journal Math.of Mathematics]] | volume=20 | year=1975 | issue=3–4 | pages=326–350 | doi = 10.1007/BF02760337 | urldoi-access=http://www.springerlink.com/content/pwh1126545520581/ | 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>

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

{{reflist}}

The modulus of convexity is known for the ''L''^''P'' spaces.<ref>{{citation

* {{cite book|author=Beauzamy, Bernard|title=Introduction to Banach Spaces and their Geometry|year=1985 [1982]|edition=Second revised|publisher=North-Holland|mr=889253|isbn=0-444-86416-4}}

| last = Hanner

| first = Olof

| title = On the uniform convexity of <math>L^p</math> and <math>\ell^p</math>

| journal = Ann.Arkiv Offör Math. (2)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.

* {{cite journal

</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 = Trans.Transactions Amer.of Math.the Soc.American Mathematical Society

| volume = 40

| year = 1936

Line 36 ⟶ 83:

| 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'', 133-175133–175, Kluwer Acad. Publ., Dordrecht, 2001. {{MR|1904276}}

* {{cite journal

| last = Day

| first = Mahlon

| title = Uniform convexity in factor and conjugate spaces

| journal = Ann. Of Math. (2)

| volume = 45

| year = 1944

| pages = 375&ndash;385

| doi = 10.2307/1969275

| issue = 2

| publisher = Annals of Mathematics

| jstor = 1969275

}}

* 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

Line 68 ⟶ 100:

| 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, 73-14973–149, 1971; ''Russian Math. Surveys'', v. 26 6, 80-15980–159.

{{Banach spaces}}

* {{cite journal | last=Pisier |first=Gilles |authorlink=Gilles Pisier | title= Martingales with values in uniformly convex spaces | journal=Israel J. Math. | volume=20 | year=1975 | issue=3–4 | pages=326–350 | doi = 10.1007/BF02760337 | url=http://www.springerlink.com/content/pwh1126545520581/ | mr=394135}}

{{Functional analysis}}

[[Category:Banach spaces]]