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.

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