Smooth and polyhedral approximation in Banach spaces

We show that norms on certain Banach spaces $X$ can be approximated uniformly, and with arbitrary precision, on bounded subsets of $X$ by $C^{\mathrm{\infty }}$ smooth norms and polyhedral norms. In particular, we show that this holds for any equivalent norm on $c_0(\mathrm{\Gamma })$, where $\mathrm{\Gamma }$ 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.