Any $\mathbb R^L$ convex set can be divided into $2^L$ number of monotonic functions Claim: The boundary of any $\mathbb R^L$ convex set, bd$(C)$, can be divided into $2^L$ hypersurfaces; each of the hypersurface can be written as the graph of a $\mathbb R^{L-1}\to\mathbb R$ monotonic function.
The claim seems trivially true. I am not sure if any theorems covering the claim or the claim is too trivial to be rigorously proved.
For a simple example, a 2D disk is a convex set; its boundary is a circle. The graph of a circle can always be divided into four (which is $2^2$) curves; each of the curves can be given by a monotonic $\mathbb R\to\mathbb R$ function. 

A monotonic function $\mathbb R^L\to\mathbb R$ is either weakly increasing or weakly decreasing in each of the directions.  
 A: I have to admit that this proof is very tedious and has no insight, and all the complex discussions just aim to avoid degenerated case.
We first prove the $2$-dimensional case. Now suppose $C\subset R^2$ is a convex set. 
Without loss of generality, we could assume this set has at least one interior point. Otherwise, it can be shown that the affine hull of $C$ is a $1$-dimensional line, which could be reduced to the $1$-dimensional case. Moreover, since for a convex set which has at least one interior point, $\text{bdry}(C)=\text{bdry}(\text{cl}(C))$, we could also assume $C$ is closed. We consider the next $2$ cases:
Case $1$: $C$ is bounded. Suppose $p=(p_x,p_y)\in \text{int} C$. Define half-plane $S=\{(x,y)| y\geq p_y\}$. Then both $S\cap \text{bdry}C$ and $S^{c}\cap \text{bdry}C$ are graph of $\textbf{proper}$ convex function.
Case $2$: $C$ is unbounded. Then $C$ has at least one direction. Without loss of generality, we could suppose:$(0,1)$ is the direction of $C$. Hence $C$ is epigraph of a convex function $f(x)$. If $f(x)$ is proper, then $C$ is the graph of $f(x)$. Otherwise, $(0,-1)$ is also the direction of $C$, in which case, either $C$ is $R^{2}$ (if $(1,0)$ and $(-1,0)$ are both directions of $C$), or $C$ is half-plane (if one of $(1,0)$ and $(-1,0)$ is direction of $C$). If both $(1,0)$ and $(-1,0)$ are not direction of $C$, then choose  $p=(p_x,p_y)\in \text{int} C$.  Suppose $p=(p_x,p_y)\in \text{int} C$. Define half-plane $S=\{(x,y)| x\geq p_x\}$. Then both $S\cap \text{bdry}C$ and $S^{c}\cap \text{bdry}C$ are graph of $\textbf{proper}$ convex function.
The remaining problem is to show that the graph of $\textbf{proper}$ convex function can be considered as the union of $2$ graph of monotonic function. This is not hard. For a proper convex function in $R^1$, either it has a global minimum, or it is monotonic. If it has a global minimum $x$, it is easy to show that $f(x)$ is monotonic in both $[x,+\infty]$ and $[-\infty,x]$.
In the general $R^n$ case, there is no difference. Just by induction, you can reduce them to the $R^2$ case. 
