  We study various regularity properties of minimizers of the
$\Phi$--perimeter, where $\Phi$ is a norm. Under suitable assumptions on $\Phi$
and on the dimension of the ambient space, we prove that the boundary of a
cartesian minimizer is locally a Lipschitz graph out of a closed singular set
of small Hausdorff dimension. Moreover, we show the following anisotropic
Bernstein-type result: any entire cartesian minimizer is the subgraph of a
monotone function depending only on one variable.
