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arXiv · 1812.03678

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

Gideon Schechtman·Nicole Tomczak--Jaegermann
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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 $\ell_\infty^m$.

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Gideon Schechtman, Nicole Tomczak--Jaegermann. 2018-12-10. Polylog dimensional subspaces of $\ell_\infty^N$. https://arxiv.org/abs/1812.03678

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