Let $f_i(x):\mathbb{R}^n\to\mathbb{R}$ be a piece-wise linear convex function with $i=1,\dots,m$. We know that $F(x)=\sum_{i=1}^m f_i(x)$ is also a piece-wise linear convex function (see this post). I am aware of the inequality $$ \sum_{i=1}^n F(x_i) \ge n \cdot F\left(\frac{\sum_{i=1}^n x_i}{n}\right), $$ that comes from Jensen's inequality . However, the only information used to derive this inequality is convexity of $F(x)$. I was wondering if better inequalities can be obtained given the information that $F(x)$ and $f_i(x)$ are convex and piece-wise linear. In the article Jensen's inequality in Wikipedia, it is mentioned that the equality of the above inequality happens if and only if $x_1=\dots=x_n$ or $F(x)$ is linear. It seems that the piece-wise linearity of $F$ (or even $f_i(x)$) could be used to obtain better bounds for $\sum_{i=1}^n F(x_i)$ than the above bound. I think similar inequalities have already been proved, but I could not find anything in the internet. What properties/inequalities do you know for such functions?
Edit: If that helps, we could consider the condition that $f_i(x):\mathbb{R}_+^n\to\mathbb{R}_+$ instead of the general one $f_i(x):\mathbb{R}^n\to\mathbb{R}$.