$\newcommand\R{\Bbb R}$Let $F$ be the set of all functions of the form $\max(a,b,c)$, where $a,b,c$ are affine functions from $\R^2$ to $\R$ and the maximum is taken pointwise. Let $G$ be the set of all functions of the form $\sum_{i=1}^n c_i f_i$, where $n$ is a natural number, the $c_i$'s are nonnegative real numbers, and the $f_i$'s are in $F$.

Is it true that any convex function from $\R^2$ to $\R$ is the pointwise limit of a sequence of functions in $G$?

$\newcommand\R{\Bbb R}$Let $F$ be the set of all functions of the form $\max(a,b,c)$, where $a,b,c$ are affine functions from $\R^2$ to $\R$ and the maximum is taken pointwise. Let $G$ be the set of all functions of the form $\sum_{i=1}^n c_i f_i$, where $n$ is a natural number, the $c_i$'s are nonnegative real numbers, and the $f_i$'s are in $F$.

Is it true that every convex function from $\R^2$ to $\R$ is the pointwise limit of a sequence of functions in $G$?