I have two functions $f:\mathbb{R}_+ \to \mathbb{R}_+$ and $g:{\mathbb{R}^k}_+ \to \mathbb{R}_+$, where $k$ is some finite natural number, $f$ is convex, increasing function and $g$ is the $L_1$ norm.
I also know that $f \circ g$ is a super-modular function. The space we are working on is a Euclidean space.
My questions are,
Is there such an $f$ ? If yes, some examples for the same is appreciated.
In addition to $f$ being convex and increasing, what more is required so that $f \circ g$ is a super-modular function?
My try: Convexity and supermodularity article suggests that super-modular and convex are somewhat equivalent.
Any help is appreciated. Thanks in advance.
First, we remark $g(x) = \sum_{i = 1}^k x_i$, since $x_i \ge 0$.
Supermodularity of $f \circ g$ requires $$f\left( \sum_{i=1}^k x_i \right) + f\left( \sum_{i=1}^k y_i \right) \le f\left( \sum_{i=1}^k \min(x_i,y_i)\right) + f\left(\sum_{i=1}^k \max(x_i,y_i)\right)$$ for all $x,y \in \mathbb R_+^k$. Hence, it is sufficient to have $$f\left( a \right) + f\left( b \right) \le f\left( c \right) + f\left(d \right)$$ for all $c \le a,b \le d$ with $a + b = c + d$. This easily follows from the convexity of $f$.