Let $C$ be the smallest closed convex cone of functions from $\mathbb{R}^n$ to $\mathbb{R}$ that contains all constant functions, all coordinate functions, and such that $\max(f,0)\in C$ whenever $f\in C$.

Is there any useful characterization of the functions lying in $C$?
It is easy to see that such functions must be convex and in addition have non-negative partial derivatives up to the second order, but I would think these conditions aren't sufficient.