Let $E$ be a Banach space. Is it possible that $E$ is super-reflexive and $\ell_p$ is crudely finitely representable in $E$ for all $p\in (1,2)$?
It seems unlikely but I cannot find an argument off the top of my head.
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Sign up to join this communityNo. If $\ell_p$ is finitely crudely representable in a Banach space $X$, then $\ell_p$ is $1+\epsilon$ finitely representable in $X$ for all $\epsilon >0$ by Krivine's theorem. You can find this in the book of Milman and Schechtman.