# On Embeddings of Finite Subsets of $\ell_p$

## TL;DR

This paper investigates the conditions under which finite subsets of _p can be embedded into Banach spaces, revealing that most such subsets embed into spaces that uniformly contain all finite-dimensional _p spaces.

## Contribution

It establishes that, aside from negligible exceptions, all finite subsets of _p embed isometrically into Banach spaces that uniformly contain _p^n for all n.

## Key findings

- Most finite subsets of _p embed into spaces containing all _p^n
- Embedding holds outside nowhere dense and Haar null sets
- Provides conditions for isometric embeddings of finite _p subsets

## Abstract

We study finite subsets of $\ell_p$ and show that, up to nowhere dense and Haar null complement, all of them embed isometrically into any Banach space that uniformly contains the spaces $\ell_p^n$, $n \in \mathbb{N}$.