Let $f$ be a non-negative function on the positive integers such that $f(s+t)\geq f(s) + f(t)$ for all $s,t\in\mathbb{Z}^+$. Consider the polytope consisting of all $x\in \mathbb{R}^n$ such that $$\sum_{i\in S} x_i \geq f(|S|)$$ for all subsets $S\subseteq \{1,\dots,n\}$. My question is: is there a compact description of the corners of this polytope? My conjecture is that the corners simply correspond to permutations $\sigma$ where one sets $x_{\sigma(1)} = f(1)$, $x_{\sigma(2)} = f(2) - f(1)$, $x_{\sigma(3)} = f(3) - f(2)$, and so on.
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