Strictly convex space: Difference between revisions

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

[[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|normed]] [[topological vector space]] (''VX'', || ||) for which the closed unit [[unitBall (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'' inon the [[boundaryunit (topology)|boundarysphere]] ∂''B'' (i.e. the [[Boundary (topology)|boundary]] of the unit ball ''B'' of ''VX''), the [[affinesegment line]] ''L''(''x'', ''y'') passing throughjoining ''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 [[Banachnormed vector space]] (''VX'',&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 [[Banachnormed vector space]] (''VX'',&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;''0<α<1''&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]] ''δ'' for (''VX'',&nbsp;||&nbsp;||) satisfies ''δ''(2)&nbsp;=&nbsp;1.

==See also==

* A [[Banach space]] (''V'',&nbsp;||&nbsp;||) is strictly convex [[if and only if]] ''x&nbsp;≠&nbsp;y'' and ''||&nbsp;x&nbsp;||=||&nbsp;y&nbsp;||=1'' → ''||&nbsp;x&nbsp;+&nbsp;y&nbsp;|| < 2''.

* [[Uniformly convex space]]

* A [[Banach space]] (''V'',&nbsp;||&nbsp;||) is strictly convex [[if and only if]] ''x&nbsp;≠&nbsp;y'' and ''||&nbsp;x&nbsp;||=||&nbsp;y&nbsp;||=1'' → ''||&nbsp;αx&nbsp;+&nbsp;(1-α)y&nbsp;|| < 1'' for all ''0<α<1''

* [[Modulus and characteristic of convexity]]

==References==

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{{Functional analysis}}

[[Category:Convex analysis]]

[[Category:Normed spaces]]

{{hyperbolic-geometry-stub}}