General context:
Let $G_1, G_2:[0,1] \rightarrow \mathbb R$ be continuous convex piece-wise affine functions such that $G_1(0)=G_2(0)=0$ and $G_1-G_2$ is not positive everywhere, i.e., their graphs cross at least once. I am interested in the set of continuous functions $F:[0,1] \rightarrow \mathbb R$ such that $G_1-F$ is still a convex function and $G_1-F\geq G_2$.
My conjecture is that $F$ can be any continuous function that lies below the convex hull of $G_1-G_2$, that is, the largest convex function under $G_1-G_2$. I am very far away from a proof and the investigation of that problem made me think about convexity and weak derivatives (in the sense of disttributions), hence my question below.
Question:
For $G_1-F$ to be convex, $F$ can be at most "as much convex" as $G_1$. A usual measure of convexity is the second derivative. Taking the functions $F$ such that $G_1'' \geq F''$ would do the trick for the first condition. But what happens when these derivative do not exist in the usual sense? Can we use other derivatives for the notion of convexity (at least in this case where we have piece-wise affine continuous functions) instead of going back to the synthetic definition of convexity?
The problem here is that $G$ is piece-wise affine. For the first derivative, it is not much of a problem, we could choose to look at the left derivative, and we would get a left-continuous step function. If we were to take the left-derivative of that step function, we loose all information since we would get $0$. But that left-continuous step function has a derivative in the distributional sense, or in the discrete sense if you want (the jumps), which will be a weighted sum of Dirac's deltas.
For instance, suppose that left derivative of $G_1$ is $-1$ on $[0,1/3)$, $-1/2$ on $[1/3,2/3]$ and $2/3$ on $[2/3,1]$. Then the distributional derivative of that function is $g_1''(x)=\frac{1}{2}1_{x=1/3}(x)+ \frac{7}{6}1_{x=2/3}(x)$. Could $g_1''$ be used in any way instead of the non-existing $G_1''$ in $G_1'' \geq F''$? For instance, could this inequality mean $1/2 \geq F''(1/3)$ and $7/6\geq F''(2/3)$, where $F''$ is the usual second derivative if it exists or the distributional second derivative?
Or maybe we could use a modification of $g_1''$? For instance the function $\tilde g_1''(x)=\frac{1}{2}1_{[1/3,2/3)}(x)+ \frac{7}{6}1_{[2/3,1]}(x)$ or the piece-wise affine function $\hat g_1''$ defined by $\hat g_1''(0)=0$, $\hat g_1''(1/3)=1/2$, $\hat g_1''(2/3)=7/6$, $\hat g_1''(1)=0$. Of course, for any modification chosen, one must apply the same modification to $F''$ when the usual second derivative does not exist.
Would any of that work to make sure $G_1-F$ is convex? If none of the above work, is there another way of exploiting the distributional derivative to achieve the goal?