I am wondering if anyone could help me figure out the following problem.
Let $F$ denote the set of all functions $f: [0,1] \to [0,1]$ such that $f$ is weakly convex, increasing, continuous (at end points), $f(0) =0$, $f(1) =1$, and falls between $\overline{f}$ and $\underline{f}$ such that $\overline{f}(x) = x$ for all $x \in [0,1]$, $\underline{f}(x) = 0$ for $x \le 1/2$, and $\underline{f}(x) = 2x-1$ for $x \ge 1/2$.
We can regard $F$ as a convex set in the set of all continuous functions from $[0,1]$ to $[0,1]$.
My question is: What would be extreme points of $F$? In particular, can we say something like any extreme point, as a function of $x$, has less than a certain number of kinks?
Thank you very much :)