Establishing convexity of a function

Question

Let $\theta \in \Theta \subset \mathbb{R}^k$. I have the following objective function $$ F(\theta):=||\max(0,f_1(\theta)),...,\max(0,f_n(\theta))||^2 $$ where $||\cdot||$ is the Euclidean Norm and $f_i(\theta): \Theta \rightarrow \mathbb{R}$ for $i=1,...,n$. Is $F(\theta)$ convex (not necessarily strictly) in $\Theta$? Or, can I exclude that $F(\theta)$ is concave in $\Theta$?

Answer 1

Without further assumptions, the only thing you can conclude is that $F$ is nonnegative. If $g$ is an arbitrary nonnegative function on $\Theta$, and all $f_i=\sqrt{g/n}$, then $F=g$.