Modulus and characteristic of convexity: Difference between revisions

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In [[mathematics]], the '''modulus and 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==

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

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

where ''BS'' denotes the closed unit ballsphere of (''X'',&nbsp;||&nbsp;||). 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>

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).

==Properties==

* The modulus of convexity, ''δ''(''ε''), is a [[monotonic function|non-decreasing]] function of ''ε''. Goebel claims the modulus of convexity is itself convex, while Lindenstrauss and Tzafriri claim that the(The modulus of convexity need not itself be a [[convex function]] of ''ε''.<ref>p. 67 in [[Lindenstrauss, Joram]]; Tzafriri, Lior, "Classical Banach spaces. II. Function spaces". ''Ergebnisse der Mathematik und ihrer Grenzgebiete'' [Results in Mathematics and Related Areas], 97. ''Springer-Verlag, Berlin-New York,'' 1979. x+243 pp.</ref>)

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

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{{reflist}}

* {{cite journal

| last = Goebel

| first = Kazimierz

| title = Convexity of balls and fixed-point theorems for mappings with nonexpansive square

| journal = Compositio Mathematica

| volume = 22

| issue = 3

| year = 1970

| pages = 269&ndash;274

}}

*{{cite book

|author=Beauzamy, Bernard

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|edition=Second revised

|publisher=North-Holland

}}

* {{cite journal

| last = Clarkson

| first = KazimierzJames

| title = Uniformly convex spaces

| journal = Trans. Amer. Math. Soc.

| volume = 2245

| year = 19701944

| pages = 269375&ndash;274385

}}

* {{cite journal

| last = GoebelDay

| first = Mahlon

| title = Uniform convexity in factor and conjugate spaces

| journal = Ann. of Math. (2)

| volume = 40

| year = 1936

| pages = 396&ndash;414

}}

* [[Joram Lindenstrauss|Lindenstrauss, Joram]] and Benyamini, Yoav. ''Geometric nonlinear functional analysis'' Colloquium publications, 48. American Mathematical Society.