In this question, all Banach spaces will be infinite-dimensional and separable, and all subspaces will be infinite-dimensional and closed.
Say that a subset of the unit sphere $S_X$ of a Banach space $X$ is asymptotic if it intersects every subspace of $X$. For $A \subseteq S_X$ and $\varepsilon > 0$, let $(A)_\varepsilon := \{x \in S_X \mid B(x, \varepsilon) \cap A \neq \varnothing\}$.
Say that the space $X$ satisfies the approximate pigeonhole principle (APH) if the following condition is satisfied:
For every asymptotic set $A \subseteq S_X$ and every $\varepsilon > 0$, there exists a subspace $Y$ of $X$ such that $S_Y \subseteq A_\varepsilon$.
It has been proved by Gowers that (APH) holds in $c_0$-saturated spaces [1]. It is a consequence of Odell and Schlumprecht's solution of the distortion problem [3] that (APH) does not hold in the $\ell_p$'s, $1 \leqslant p < \infty$. Milman showed that a Banach space not containing isomorphic copies of $\ell_p$, $1 \leqslant p < \infty$, or $c_0$, has a distortable subspace, hence not satisfying (APH) [3].
Two times recently, a convex version of (APH) naturally appeared in my research. Say that $X$ satisfies (CAPH) if the following holds:
For every asymptotic set $A \subseteq S_X$ and every $\varepsilon > 0$, there exists a subspace $Y$ of $X$ such that $Y \cap \overline{conv}((A)_\varepsilon)$ has nonempty interior in $Y$.
(Here, $\overline{conv}$ denotes the closed convex hull.)
Obviously (APH) is stronger than (CAPH), so I was wondering whether, by any chance, (CAPH) could hold for more Banach spaces than essentially just $c_0$.
Question 1: Are there examples of Banach spaces satisfying (CAPH) but not (APH)?
Question 2: Are there Banach spaces failing (CAPH)?
I'm mostly interested in Question 2. I'm thinking that maybe a counterexample could follow from known proofs about distortion, but I'm really not a expert in that so I wouldn't see it myself...
