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...

  • $\begingroup$ Noe: Very interesting questions. If I understand correctly spaces that fail CARH should be (probably) be distortable. The space for which is easiest to see that it is 2-distortable is Tsirelson's space (only about a one page proof). Have you looked there? There is also my paper with Ryan and Pavlos containing spaces that are arbitrarily distortable because they have c_0 and \ell_1 blocks in every subspace. $\endgroup$ Sep 9, 2021 at 14:58
  • $\begingroup$ Hi @KevinBeanland and thank you for your comment! What I know for sure is that uniformily convex spaces failing CAPH have distortable subspaces; I don't know if CAPH => distortability in general but I guess that indeed distortable spaces are natural to look at. I'll have a look at the two examples you're telling me about, maybe p-convexifications of the Tsirelson also could be a good starting point. $\endgroup$ Sep 13, 2021 at 17:30


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