  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").
