I think the answer is no. Indeed, consider the case when $n=2$ and $f$ is the pointwise supremum of the set of all affine functions $g$ such that $g(0,0)\le-1$, $g(0,h)\le-1+2h$, $g(0,-h)\le-1+2h$, $g(1,h)\le0$, $g(1,-h)\le0$, $g(-1,h)\le0$, and $g(-1,-h)\le0$, where $h>0$ is small enough.