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Let $H$ be a Hilbert space and $C$ be a non empty closed convex subset of $H$ and let $x\notin C$. We know that there exists a unique $y_0$ in $C$ such that $\|x-y_0\|=\inf_{y\in C}\|x-y\|$. Call $y_0$, the projection of $x$ onto $C$.

The proof of this result heavily depends on the parallelogram law which holds only in Hilbert spaces. Is the result true for just normed spaces also? Have people already studied about this?

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    @Ashok The vector the existence of which is assured in that lemma has some properties which resemble somewhat those of the $y_0$ you are reffering to - such a $y_0$ need not exist in general. It depends on what you are up to whether you can use it or not.2011-12-29

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For Banach spaces, see the links (the MO link in particular) in my comments for positive results: A uniformly convex Banach space $X$ has your property and a Banach space with your property is reflexive. In the MO post, a link to a space with your property that is not uniformly convex is given.

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Because you ask whether it is true for just normed spaces, with no mention of completeness, I will mention that it is false for every incomplete inner product space. Suppose $X$ is an incomplete inner product space, and let $\overline X$ be its (Hilbert space) completion. Let $y$ be an element of $\overline X\setminus X$, and let $C=\{x\in X:\|x-y\|\leq \frac{1}{2}\|y\|\}$. Then $C\subset X$ is closed, convex, and nonempty, but has no element of smallest norm.

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    Thanks Jonas. Nice example.2011-12-31