Define function $f^i := h^i - g^i$, where $f^i$, $g^i$ are real-valued convex functions for all $i = 1,\dots, 10$. Do there exist real-valued convex functions $h$, $g$ where $h - g = \max\limits_{i} f^i$?
I don't even know how to even approach this problem.
On
Here is a closed form expression to write the finite maximum of DC functions as a DC function:
$$\max_{i=1,\ldots,n}\{h^i - g^i\} = \underbrace{\max_{i=1,\ldots,n}\Big\{h^i + \sum_{k=1,k \ne i}^n g^k\Big\}}_{F_1} - \underbrace{\sum_{i=1}^n g^i}_{F_2}$$ where the functions $F_1$ and $F_2$ on the right hand side are convex given that $h^i$ and $g^i$ are convex functions for all $i=1,\ldots,n$.
I learned it from Proposition 3.2 of On DC and Local DC Functions by Liam Jemison.
Yes, the class of DC functions (difference-convex, or delta-convex as they are sometimes called) is a lattice: it's closed under taking pointwise max or min. It suffices to show that for any two DC functions $f_1, f_2$ their maximum $\max(f_1, f_2)$ is also DC.
Note that $\max(f_1, f_2) = \frac12|f_1+f_2| + \frac12|f_1-f_2|$, where both $f_1 \pm f_2 $ are DC. Thus, it suffices to show that the absolute value of a DC function is DC.
If $f$ is DC, then from $f = h-g$ we get $\max(f, 0) = \max(h, g) - g$, hence $\max(f, 0)$ is also DC.
And then $|f| = \max(f, 0) + \max(-f, 0)$ is DC as well.
This proof, and much more, can be found in On difference convexity of locally Lipschitz functions by Miroslav Bačak and Jonathan M. Borwein.