# Polylog dimensional subspaces of $\ell_\infty^N$

## TL;DR

This paper demonstrates that high-dimensional subspaces of _^N contain large nearly-isometric copies of _^k, with the size of these copies depending on the dimension and the ambient space.

## Contribution

It establishes new bounds on the size of nearly-isometric _^k subspaces within high-dimensional subspaces of _^N, advancing understanding of their geometric structure.

## Key findings

- Subspaces of dimension > (\u221f N    )^2 contain _^k copies with k.
- Any subspace of dimension n contains a subspace of dimension proportional to 
()
() with distance at most 1+.
- The results depend on parameters  and , providing bounds on the size and distortion of _^k subspaces.

## Abstract

We show that a subspace of of $\ell_\infty^N$ of dimension $n>(\log N\log \log N)^2$ contains $2$-isomorphic copies of $\ell_\infty^k$ where $k$ tends to infinity with $n/(\log N\log \log N)^2$. More precisely, for every $\eta>0$, we show that any subspace of $\ell_\infty^N$ of dimension $n$ contains a subspace of dimension $m=c(\eta)\sqrt{n}/(\log N\log \log N)$ of distance at most $1+\eta$ from $\ell_\infty^m$.