# Convex pigeonhole principle in Banach spaces

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...

• 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. Sep 9, 2021 at 14:58
• 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. Sep 13, 2021 at 17:30