  We show that norms on certain Banach spaces $X$ can be approximated
uniformly, and with arbitrary precision, on bounded subsets of $X$ by
$C^{\infty}$ smooth norms and polyhedral norms. In particular, we show that
this holds for any equivalent norm on $c_0(\Gamma)$, where $\Gamma$ is an
arbitrary set. We also give a necessary condition for the existence of a
polyhedral norm on a weakly compactly generated Banach space, which extends a
well-known result of Fonf.
