Let $Y$ be a Banach space isomorphic to $\ell\_p$, $1<p<\infty$. Is it true that any finite subset of $\ell\_p$ is isometric to some finite subset of $Y$?


It seems to me that it is an interesting question. It can be regarded as a special case of a recent question [Under which conditions is it possible to find points with same distances under bi-Lipschitz map](https://mathoverflow.net/questions/226446/) (which was closed) and is related to my (unanswered) question [Isometric embeddings of finite subsets of $\ell\_2$ into infinite-dimensional Banach spaces](https://mathoverflow.net/questions/221181/)


In connection with this question it is worth mentioning that there
is a theory developed by Krivine in Ann. Math. (2) 104, 1-29
(1976) (with important additions by Maurey and Pisier (Stud. Math.
58, 45-90 (1976)) and further simplifications by other authors,
see Chapter 12 in the book by Benyamini-Lindenstrauss on Geometric
Nonlinear Functional Analysis or Part II in the book
Milman-Schechtman on Asymptotic Theory) which implies that for any
$\varepsilon>0$ any finite-dimensional subspace of $\ell\_p$ embeds
into any Banach space isomorphic to $\ell\_p$ with distortion $\le
(1+\varepsilon)$.


Added on 4/3/2017: In a [recent paper](https://arxiv.org/pdf/1704.00319.pdf) James Kilbane proved that the set of possible counterexamples (if they exist) is small in a certain sense.



Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?
We (me and the author posing the question) have answered this question in the negative - for each $p \neq 2$ we have found a space $X$ such that $X$ is isomorphic to $\ell\_p$ but there are 5 element subsets of $\ell\_p$ that do not embed isometrically into $X$. Our paper is on arxiv here: <https://arxiv.org/abs/1708.01570>


The question about $\ell\_2$ is still open.


