  For suitable bounded hyperconvex sets $\Omega$ in $\mathbb{C}^N$, in
particular the ball or the polydisk, we give estimates for the approximation
numbers of composition operators $C_\phi \colon H^2 (\Omega) \to H^2 (\Omega)$
when $\phi (\Omega)$ is relatively compact in $\Omega$, involving the
Monge-Amp\`ere capacity of $\phi (\Omega)$.
