[Math] Best approximation for a normed vector space $X$

Question

I am self-studying functional analysis. As far as I know, a best approximation of $X$ by a closed subspace $C \subseteq X$ exists and is unique if $X$ is a Hilbert space, a uniformly convex Banach space, or more generally, a reflexive space.

My question is: is there a even more general condition on the space $X$ for the existence and uniqueness of the best approximation of $X$ by a closed and convex subspace $C \subseteq X$?

Any comments or suggested readings are welcome.

Answer 1

Sharper results: Best Approximation by Closed Sets in Banach Spaces and On a sufficient condition for proximity by Ka Sing Lau. Quote from the first:

The set $K$ is called proximinal (Chebyshev) if every point $x\in X$ has a (unique) best approximation from $X$. lt is easy to see that every closed convex set $X$ in a reflexive space $X$ is proximinal. In addition, if the norm is strictly convex, then $X$ is Chebyshev. However, if $X$ is not assumed reflexive or $X$ is not assumed convex, then the above result is false in general. In [7] Steckin introduced the concept of almost Chebyshev. A set $K$ is called almost Chebyshev if the set of $x$ in $X$ such that $K$ fails to have unique best approximation to $x$ is a first category subset of $X$. He proved that if $X$ is a uniformly convex Banach space, then every closed subset is almost Chebyshev. By using this concept, Garkavi [4] showed that for any reflexive subspace $F$ in a separable Banach space, there exists a (in fact, many) subspace $G$ which is $B$-isomorphic to $F$ and is almost Chebyshev. The author [6] showed that if $X$ is a separable Banach space which is locally uniformly convex or possesses the Radon-Nikodym property, then "almost all" closed subspaces are almost Chebyshev. In [3], Edelstein proved that if $X$ has the Radon-Nikodym property, then for any bounded cIosed convex subset $K$, the set of $x$ in $X$ which admit best approximations from $K$ is a weakly dense subset in $X$.