4 problems

Let $S\in\{0,1\}^{m\times n}$. A matrix is a slack matrix if it is the slack matrix of a polytope. Polynomial-time recognition conjecture. There is an algorithm polynomial in $m,n$…

Let $S_n$ be the slack matrix of the regular $n$-gon, and let $\operatorname{rank}_{\operatorname{psd}}(S_n)$ denote its positive semidefinite rank. Logarithmic psd-rank conjecture…

Homogeneous boolean rank conjecture. For any $n$,

Let $P$ be a psd-minimal $d$-polytope, and let its support matrix be the $0,1$ matrix with the same zero pattern as a slack matrix of $P$. Support-matrix rank conjecture. The suppo…