Strictly convex space: Difference between revisions

Revision as of 19:31, 12 August 2007

Latest revision as of 02:22, 5 October 2023

(28 intermediate revisions by 20 users not shown)

Line 1:

{{Short description|Normed vector space for which the closed unit ball is strictly convex}}

In [[mathematics]], a '''strictly convex space''' is a [[normed vector space|normed]] [[topological vector space]] (''V'', || ||) for which the [[unit ball]] is a strictly [[convex set]]. Put another way, a strictly convex space is one for which, given any two points ''x'' and ''y'' in the [[boundary (topology)|boundary]] ∂''B'' of the unit ball ''B'' of ''V'', the [[affine line]] ''L''(''x'', ''y'') passing through ''x'' and ''y'' meets ∂''B'' ''only'' at ''x'' and ''y''.

[[Image:Vector norms.svg|frame|right|The unit ball in the middle figure is strictly convex, while the other two balls are not (they contain a line segment as part of their boundary).]]

In [[mathematics]], a '''strictly convex space''' is a [[normed vector space]] (''X'', || ||) for which the closed unit [[Ball (mathematics)|ball]] is a strictly [[convex set]]. Put another way, a strictly convex space is one for which, given any two distinct points ''x'' and ''y'' on the [[unit sphere]] ∂''B'' (i.e. the [[Boundary (topology)|boundary]] of the unit ball ''B'' of ''X''), the segment joining ''x'' and ''y'' meets ∂''B'' ''only'' at ''x'' and ''y''. Strict convexity is somewhere between an [[inner product space]] (all inner product spaces being strictly convex) and a general [[normed space]] in terms of structure. It also guarantees the uniqueness of a best approximation to an element in ''X'' (strictly convex) out of a convex subspace ''Y'', provided that such an approximation exists.

If the normed space ''X'' is [[Banach space|complete]] and satisfies the slightly stronger property of being [[Uniformly convex space|uniformly convex]] (which implies strict convexity), then it is also reflexive by [[Milman–Pettis theorem]].

==Properties==

The following properties are equivalent to strict convexity.

* A [[Banach space]] (''V'', || ||) is strictly convex [[if and only if]] the [[modulus of convexity]] ''δ'' for (''V'', || ||) satisfies ''δ''(2) = 1.

* A [[normed vector space]] (''X'',&nbsp;||&nbsp;||) is strictly convex if and only if ''x''&nbsp;≠&nbsp;''y'' and ||&nbsp;''x''&nbsp;||&nbsp;=&nbsp;||&nbsp;''y''&nbsp;||&nbsp;=&nbsp;1 together imply that ||&nbsp;''x''&nbsp;+&nbsp;''y''&nbsp;||&nbsp;<&nbsp;2.

* A [[normed vector space]] (''X'',&nbsp;||&nbsp;||) is strictly convex if and only if ''x''&nbsp;≠&nbsp;''y'' and ||&nbsp;''x''&nbsp;||&nbsp;=&nbsp;||&nbsp;''y''&nbsp;||&nbsp;=&nbsp;1 together imply that ||&nbsp;''αx''&nbsp;+&nbsp;(1&nbsp;&minus;&nbsp;''α'')''y''&nbsp;||&nbsp;&lt;&nbsp;1 for all 0&nbsp;&lt;&nbsp;''α''&nbsp;&lt;&nbsp;1.

* A [[normed vector space]] (''X'',&nbsp;||&nbsp;||) is strictly convex if and only if ''x''&nbsp;≠&nbsp;''0'' and ''y''&nbsp;≠&nbsp;''0'' and ||&nbsp;''x''&nbsp;+&nbsp;''y''&nbsp;||&nbsp;=&nbsp;||&nbsp;''x''&nbsp;||&nbsp;+&nbsp;||&nbsp;''y''&nbsp;|| together imply that ''x'' = ''cy'' for some constant ''c&nbsp;>&nbsp;0'';

* A [[Banachnormed vector space]] (''VX'',&nbsp;||&nbsp;||) is strictly convex [[if and only if]] the [[modulus of convexity]] ''&delta;δ'' for (''VX'',&nbsp;||&nbsp;||) satisfies ''&delta;δ''(2)&nbsp;=&nbsp;1.

==See also==

* [[Uniformly convex space]]

* [[Modulus and characteristic of convexity]]

==References==

* {{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

}}

{{Functional analysis}}

[[Category:Convex analysis]]

[[Category:Normed spaces]]

{{hyperbolic-geometry-stub}}