On embeddings of finite subsets of $\ell_2$

TL;DR

This paper investigates the conditions under which finite subsets of  can be isometrically embedded into Banach spaces, addressing a question about the universality of infinite-dimensional Banach spaces for such subsets.

Contribution

It provides partial answers to whether every infinite-dimensional Banach space contains all finite subsets of  isometrically, and discusses related embedding properties.

Findings

Some finite subsets of  do not embed isometrically into certain Banach spaces.

Theorem 3.1, related to embedding properties, was previously proven by Shkarin.

Abstract

We study finite subsets of $\ell_2$, 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 $\ell_2$ isometrically. The updated version contains acknowledgement that Theorem 3.1 has been proven previously in a paper of Shkarin.