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

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").