  Let $n\geq C$ for a large universal constant $C>0$, and let $B$ be a convex
body in $R^n$ such that for any $(x_1,x_2,\dots,x_n)\in B$, any choice of signs
$\varepsilon_1,\varepsilon_2,\dots,\varepsilon_n\in\{-1,1\}$ and for any
permutation $\sigma$ on $n$ elements we have
$(\varepsilon_1x_{\sigma(1)},\varepsilon_2x_{\sigma(2)},\dots,\varepsilon_nx_{\sigma(n)})\in
B$. We show that if $B$ is not a cube then $B$ can be illuminated by strictly
less than $2^n$ sources of light. This confirms the Hadwiger--Gohberg--Markus
illumination conjecture for unit balls of $1$-symmetric norms in $R^n$ for all
sufficiently large $n$.
