A distance-minimizing continuous projection onto a finite-dimensional subspace?

Question

Let $E$ be a Banach space, which need not be a Hilbert space, and let $F$ be a finite-dimensional subspace of $E$. Suppose that for all $x \in E$, there exists a $y \in F$ realizing the minimal distance of $x$ to $F$. Does this imply that there is a continuous projection $p : E \rightarrow F$ minimizing the distance?

Answer 1

$\newcommand{\P}{\mathscr{P}}$Let $E$ be a normed vector space, and let $F$ be a subset. For $x \in E$, let $$ \P_F(x) = \left\{ x_0 \in E \mid \|x-x_0\| = \inf_{y \in E} \|x-y\| \right\}. $$ Now, suppose that $F$ is proximinal, i.e., that $\P_F(x) \neq \emptyset$ for all $x \in E$. Then a metric selection of $E$ is a function $\pi : E \to F$ such that $\pi(x) \in \P_F(x)$ for all $x \in E$. If I have now correctly understood your question, what it amounts to is the following:

Let $E$ be a Banach space, and let $F$ be a finite-dimensional proximinal subspace of $E$. Does there exist a continuous metric selection $\pi : E \to F$ of $F$?

Poking about the literature yields the following characterisation of the existence of a continuous metric selection, for what it's worth.

[Deutsch–Li–Park, 1989] Let $E$ be a Banach space, and let $F$ be a proximinal subspace of $E$. Let $\ker \P_F := \{x \in E \mid \P_F(x) = \{0\}\}$. Then the following are equivalent:

1. $F$ admits a continuous metric selection.
2. $F$ admits a continuous metric selection which is homogeneous and additive modulo $F$.
3. $\ker \P_F$ contains a closed homogeneous subset $N$ such that the canonical map $\omega : E \to E/F$ restricts to a homeomorphism $\left.\omega\right|_N : N \to E/F$.

In general, however, continuous metric selections need not exist:

[Lazar–Wulbert–Morris, 1969] Let $E = L^1(X,\mu)$ for $(X,\mu)$ a non-atomic positive measure space. Then for any finite-dimensional (and hence necessarily proximinal) subspace $F$ of $E$, there does not exist a continuous metric selection.

In general, there's a fair bit of literature on the existence of continuous metric selections, particularly for finite-dimensional proximinal subspaces, but a cursory look suggests that there might not be all that much in the way of general theory. The moment, however, that $F$ is Chebyshev (i.e., $\P_F(x)$ is a singleton for all $x \in E$), for instance when $E$ is reflexive and strictly convex, then the unique metric selection, the metric projection, is necessarily continuous.