Polylog dimensional subspaces of ${\ell }_{\infty }^N$

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 ${\ell }_{\infty }^m$.