Extremal Lipschitz convex functions Let $B_d$ the unit ball in $\mathbb{R}^d$, and let $F_d$ be the set of convex functions with Lipschitz constant at most 1 from $B_d$ to $\mathbb{R}$. 
When $d=1$ (so the domain is the just the interval $[-1,1]$), we can write every function $f\in F_1$ as a convex combination of very "simple" functions of the form $g_y(x) = |y-x|$ (up to an additive constant). That is, there is a distribution $P$ on $[-1,1]$ and a $c\in\mathbb{R}$ such that, for all $x$, we have $f(x) = E_{y\sim P} (g_y(x))+c$. The absolute value functions $g_y$ are thus the extremal points of $F_1$. (One can get rid of the constant $c$ by restricting $F_1$ to functions such that $f(0)=0$ and adjusting the $g_y$'s accordingly.) 
My question is: what are the extremal points of $F_d$? Is there a simple collection of such functions whose convex hull is (dense in) the whole space? Is that set unique? I have a feeling this is well known, but could not figure out where to look for the result. 
 A: A bit is known about the extreme points of the set $\{f: B_d \to \mathbb{R}\, |\, f(0) = 0$ and $L(f) \leq 1\}$ where $L(f)$ denotes the Lipschitz number of $f$. But you are asking for the extreme points of the family of convex functions in this set. I don't know of any previous work on this.
One thing we can say is that any function satisfying $|\nabla f| = 1$ almost everywhere is an extreme point of the first set. (And such functions can be fairly complicated.) So if it is also convex then it must be an extreme point of the second, smaller set. This would cover your shifted absolute value functions in the $d = 1$ case, for example. An obvious higher-dimensional analog would be the shifts of the functions $f(x) = |x - y|$ for $y \in B_d$.
Some notation: for any function $f$, let $\tilde{f} = f - f(0)$ be the shifted function which satisfies $\tilde{f}(0) = 0$. Then it seems to me that in the $d = 2$ case, if $L \subset B_2$ is a line segment and $f(x) = \rho(x,L)$ where $\rho$ is euclidean distance, then $|\nabla f| = 1$ almost everywhere and $f$ is convex, so $\tilde{f}$ is a function of your type. I guess this generalizes for arbitrary $d$ to the functions $f(x) = \rho(x,L)$ where $L \subset B_d$ is a $\leq d-1$-dimensional convex subset.
I think it's a good conjecture that the convex hull of these functions is dense in the whole set, but I don't know this.
