Do you know any result concerning the representation of functions $f : [0,1] \to [0,1]$ continuous and increasing (with $f(0)=0,f(1)=1)$) as convex combinations of a family of particular functions?
If we denote by $\mathcal{K}$ the family of functions like above the question is if there exists a family $\mathcal{B}$ (strictly smaller than $\mathcal{K}$) of functions as above such that each $f \in \mathcal{K}$ can be written as $$ \sum_{i \in I}\alpha_i b_i \text{ with } b_i \in \mathcal{B} \text{ and } \sum_{i\in I}\alpha_i=1 (\alpha_i>0) $$
I think a related question is the following:
If we denote by $M=\{f:[0,1] \to [0,1]: f(x)= x^\lambda , \lambda > 0\}$ then what is the convex hull of $M$? Is it close to the family $\mathcal{K}$ above? I guess it cannot be equal, since curves which simultaneously have low slope around zero and low slopes around one are not in $M$.
If we view $\mathcal{K}$ as a subset of $C([0,1])$ we can easily see that it is convex. Is there any way we can impose some restrictions on $\mathcal{K}$ ($C^1$ with bounded derivatives, Lipschitz, etc) so that the resulting set is still convex and has reasonable extremal points (in order to use Krein Milman theorem)?
The subset of piecewise linear functions is such a subset. Take an increasing piece wise linear function $f$ that fixes 0 and 1 such that $1/2 f$ stays within 1/2 of the function while remaining below it (such as being 0 until the function is close to 1/2, then rising to 1/2 until the end). Then take another piece wise linear function $g$ such that $1/4 g+1/2 f$ stays within 1/4 of the function, while still remaining below it. Continuing in this way, we get the original function as a convex combination of piece wise linear functions.
A certain subset of increasing piece wise linear functions fixing the end points is called Thompson's group.
Edit As pointed out in the comments, this doesn't work. This construction works for $C^1$ functions, and possibly Lipschitz or absolutely continuous functions.