Consider a ground set $U=\{1,2,...,n\}$ and the set $\mathcal{F}$ of all non-negative submodular functions $f: 2^U \mapsto \mathbb{R}_{\geq 0}$. The set $\mathcal{F}$ is closed under taking linear combinations of elements but with non-negative coefficients, and hence it forms a cone. Question: does this cone have $O(2^{poly(n)})$ number of extreme rays?In other words, does there exist a set $\mathcal{B} \subseteq \mathcal{F}$ of non-negative submodular functions over $U$ such that any non-negative submodular function $f$ over $U$ can be represented as $f = \sum_{b\in \mathcal{B}}  \gamma_b \cdot b$ for $\gamma_b \in \mathbb{R}_{\geq 0}$, and such that the size of $\mathcal{B}$ is exponential in $n$?I don't need a characterization of set $\mathcal{B}$, just quantify its size.I came across this paper https://www.jstor.org/stable/1997813?seq=21#page_scan_tab_contents which in Section 2.1 states that such a cone has just $O(2^n\cdot n^2)$ facets, because we can define such a cone as an intersection of $O(2^n\cdot n^2)$ halfspaces each defined for a different triple $(X ,x,y) \subseteq 2^U \cdot U \cdot U$ by $f(X+x)+f(X+y)-f(X + x + y) -f(X) \geq 0$. I have a hard time reading the paper, so can someone tell me if this argument is already answering my question?I will greatly appreciate any help. EDIT: Would it change a lot, if we would be considering only monotone submodular functions? 
Extreme rays of the cone of submodular functions
submodularity