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Convex hull of all-ones principal submatrices

For a subset $S$ of $\{1,\ldots,n\}$, let $\mathbf{1}_S\in\{0,1\}^n$ denote the indicator vector of $S$, with a $1$ on the $i$th coordinate iff $i\in S$. Let $\mathcal{X}$ denote the convex-hull of al …