Consider $\mathbb{R}^n$ with $\ell_{\infty}$ norm. Let $Y \subset \mathbb{R}^n$ be a linear subspace with dimension $r < n$.
I am considering the following projection: for each $x \in \mathbb{R}^n$, $P(x) := \text{argmin}_{y \in Y} ||y - x ||_{\infty}$.
I know that it may not be well defined because there might be multiple $y$'s to achieve the minimum. If this is the case, I can choose one of them which has the minimum $\ell_2$ norm, i.e. $P(x) := \{y \in \text{argmin}_{y \in Y} ||y - x ||_{\infty} : \forall y' \in \text{argmin}_{y \in Y} ||y - x ||_{\infty}, ||y||_2 \leq ||y'||\}.$
Then, I think it is well-defined since $\ell_2$ norm is strictly convex. Also, I believe that this projection is continuous with respect to $\ell_{\infty}$ norm.
Let's define \begin{equation*} ||P||_{\infty} := \sup_{||X||_{\infty} \leq 1} ||P(X)||_{\infty}. \end{equation*}
My question is, is $P$ non-expansive? In other words, is it true that $||P||_{\infty} \leq 1$?
I googled this and only found that Bound on projection to finite dimensional subspace of Banach Space .
But, what I really want is the non-expansiveness. Is it false in general?
Thanks.