Modulus and characteristic of convexity: Difference between revisions

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

These notions are implicit in the general study of uniform convexity by J.&nbsp;A.&nbsp;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.&nbsp;M.&nbsp;Day (.<ref>{{harvtxt|Day|1944}}).citation

| 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

}}</ref>

==Properties==

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

* The normed space {{nowrap|(''X'', ǁ&nbspthinsp;||&nbspsdot;||&thinsp;ǁ)}} is [[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}}.

* 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 {{harvtxt|Pisier|1975}}.</ref> Namely, there exists {{nowrap|''q'' &ge; 2}} and a constant&nbsp;{{nowrap|''c'' &gt; 0}} such thatcitation

| 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}}

.</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>

Line 44 ⟶ 59:

| publisher = American Mathematical Society

| jstor = 1989630

}}

*{{citation

| 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}}

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}}.

* [[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.

*{{citation

| 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}}

[[Category:Banach spaces]]