Is best approximation from a linear subspace a linear map?

Question

Let $X$ be a strictly convex Banach space, and $Y \subset X$ a closed subspace. Then for any $x \in X$ there exists a unique $y \in Y$ that minimizes the distance to $x$, i.e. a best approximation of $x$ from $Y$.

Is the mapping $x \mapsto y$ linear? Is it bounded?

Edit:

The article by F. Deutsch, Linear selections for the metric projection, Journal of Functional Analysis, Volume 49, Issue 3, December 1982, Pages 269–292, provides good answers to this question.

Answer 1

If the Banach space is a Hilbert space, it's linear, continuous and has norm 1 unless it is the zero mapping. The property you speak of is called proximinality. If your space is reflexive, it is proximinal. If not, you are likely out of luck. This map is continuous if the space is uniformly convex.