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 (which was closed) and is related to my (unanswered) question Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces

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 James Kilbane proved that the set of possible counterexamples (if they exist) is small in a certain sense.

  • $\begingroup$ Is $p$ any number in $[1,\infty]$? Also I understand that $\ell^p$ means $\ell^p(\mathbf{N})$. $\endgroup$
    – YCor
    Commented Dec 27, 2015 at 12:46
  • $\begingroup$ @YCor By $\ell_p$ I meant $\ell_p(\mathbb{N})$, but I meant $p\in(1,\infty)$ only (I corrected this). For $p=1,\infty$ one can easily answer the question using the fact that $\ell_1$ and $\ell_\infty$ are isomorphic to spaces having the uniqueness of geodesics property. Thank you for your question. $\endgroup$ Commented Dec 27, 2015 at 16:30
  • $\begingroup$ @MichailOstrovskii: For the benefit of the non experts you should add that $\ell_p$ is $1+\epsilon$ finitely representable in every isomorphic of $\ell_p$ for every $\epsilon >0$. $\endgroup$ Commented Dec 27, 2015 at 18:59
  • $\begingroup$ @BillJohnson: I added such information, thank you for the comment $\endgroup$ Commented Dec 27, 2015 at 19:40

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.