It is a know fact that in any real Hilbert space $E$, given a point $a \in E$ and $C$ a closed convex and non-empty set in $E$, theres one and only one $x \in C$ such that $d(a,x) = d(a,C)$.
My question is: can it be extended to norms not coming from an inner product in a Banach space? It is easy to prove that:
- Uniqueness of such a $x$ is guaranteed when $(E, \| . \|)$ is a strictly convex space.
- The existence always holds if $E$ is finite dimensional.
So does the existence holds without any assumption on the dimension of $E$, even if it's a strictly convex Banach space?
