Projection exists => Uniformly convex? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T08:35:04Z http://mathoverflow.net/feeds/question/81979 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81979/projection-exists-uniformly-convex Projection exists => Uniformly convex? Thomas Kuhn 2011-11-27T02:24:47Z 2011-11-29T00:12:39Z <p>Hello,</p> <p>I know that: Let X be a uniformly convex Banach-Space, $a\in X$ and $C\subset X$ closed and convex, then there is a unique $b\in C$ with $\left\Vert a-b\right\Vert=\inf_{x\in C}\left\Vert a-x \right\Vert$.</p> <p>Moreover I know that: Let X be a Banach-Space, s.t. for every $a\in X$ and $C\subset X$ closed and convex there is at most one $b\in C$ with $\left\Vert a-b\right\Vert=\inf_{x\in C}\left\Vert a-x \right\Vert$. Then X is striclty convex.</p> <p>So I wonder, if the following statement is true: Let X be a Banach-Space, s.t. for every $a\in X$ and $C\subset X$ closed and convex, there is a unique $b\in C$ with $\left\Vert a-b\right\Vert=\inf_{x\in C}\left\Vert a-x \right\Vert$. Then X is uniformly convex.</p> <p>EDIT: This statement is false, see Hsueh-Yung Lin's comment. So I should ask: Let X be a Banach-Space, s.t. for every $a\in X$ and $C\subset X$ closed and convex, there is a unique $b\in C$ with $\left\Vert a-b\right\Vert=\inf_{x\in C}\left\Vert a-x \right\Vert$. Then every bounded sequence has a weakly convergent subsequence.</p> <p>Best regards,</p> http://mathoverflow.net/questions/81979/projection-exists-uniformly-convex/82018#82018 Answer by Bill Johnson for Projection exists => Uniformly convex? Bill Johnson 2011-11-27T17:35:54Z 2011-11-27T17:35:54Z <p>Your modified question has an affirmative answer. An equivalent form, in view of the Eberlein-Smulian theorem, is whether the Banach space $X$ must be reflexive if every closed bounded non empty set admits best approximations. If $X$ is not reflexive, then by R. C. James' famous characterization of reflexivity, there is a norm one linear functional $F$ on $X$ s.t. $F$ does not achieve its norm on the closed unit ball <code>$B_X$</code>. Let <code>$C:= [F=1]\cap 2B_X$</code>. This closed bounded non empty convex set contains no point of minimal norm.</p>