Let $X$ be a separable Banach space and $D\subseteq X$ be a

• proper, connected, and dense $G_{\delta}$ subset of $X$,

• $X-D$ is $\sigma$-porous.

Then is $X-D$ contained in a finite-dimensional subspace $E$ of $X$?

**This seems at least plausible since Corollary 3.4 of this paper shows that the codimension of a prous set in (finite-dimensions) is less than the entire space... **

Let $X$ be a separable Banach space and $D\subseteq X$ be a

• proper, connected, and dense $G_{\delta}$ subset of $X$,

• $X-D$ is $\sigma$-porous.

Then is $X-D$ contained in a finite-dimensional subspace $E$ of $X$?

**This seems at least plausible since Corollary 3.4 of this paper shows that the codimension of a prous set in (finite-dimensions) is less than the entire space... **

Background/Definition(s)

• σ-porous: A set $A$ is $\sigma$-porous if it can be covered by a countable number of porous sets.
• porous: A set $A$ is porous if for each $a \in A$, every $\lambda\in (0,1)$, and every $\epsilon>0$, there exists some $b \in X-A$ satisfying $$ d(a,b)< ε \mbox{ and } B(b,\lambda d(a,b)) \cap A=\emptyset. $$
• Facts: I can be shown that if $X=\mathbb{R}^n$ then any porous set is of Lebesgue measure $0$, Haar-null, and nowhere dense. However, it can be shown that (even in this finite-dimensional setting) there exist sets which are either Lebesgue measure $0$, Haar-null, or nowhere dense but fail to be $\sigma$-porous.

Let $X$ be a separable Banach space and $D\subseteq X$ be a

• proper, connected, and dense $G_{\delta}$ subset of $X$,

• $X-D$ is $\sigma$-porous.

Then is $X-D$ contained in a finite-dimensional subspace $E$ of $X$?

Let $X$ be a separable Banach space and $D\subseteq X$ be a

• proper, connected, and dense $G_{\delta}$ subset of $X$,

• $X-D$ is $\sigma$-porous.

Then is $X-D$ contained in a finite-dimensional subspace $E$ of $X$?

**This seems at least plausible since Corollary 3.4 of this paper shows that the codimension of a prous set in (finite-dimensions) is less than the entire space... **