projections in normed linear spaces 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?

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.
