A weak version of the $\varepsilon$-Dvoretzky conjecture for normed spaces

TL;DR

This paper proves a weaker form of the $ ext{ε}$-Dvoretzky conjecture, demonstrating that high-dimensional normed spaces contain subspaces where the norm approximates a symmetric norm, with dimension proportional to logarithmic functions of n and ε.

Contribution

The authors establish a partial version of the $ ext{ε}$-Dvoretzky conjecture, showing existence of nearly symmetric subspaces of significant dimension in normed spaces.

Findings

Existence of subspaces with dimension ≥ c log n / |log ε|

Norms in these subspaces are ε-close to 1-unconditional norms

Provides a lower bound on the dimension of such subspaces

Abstract

We prove a weak version of the $\varepsilon$-Dvoretzky conjecture for normed spaces, showing the existence of a subspace of $\mathbb{R}^n$ of dimension at least $c \log n / |\log \varepsilon|$ in which the given norm is $\varepsilon$-close to a norm obeying a large discrete group of symmetries ("$1$-unconditional norm").