On embeddings of finite subsets of ${\mathrm{\ell }}_2$

We study finite subsets of ${\mathrm{\ell }}_2$, and more generally any metric space, and consider whether these isometrically embed into a Banach space. Our results partially answer a question of Ostrovskii, on whether every infinite-dimensional Banach space contains every finite subset of ${\mathrm{\ell }}_2$ isometrically. The updated version contains acknowledgement that Theorem 3.1 has been proven previously in a paper of Shkarin.