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

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