Inscribability of a polytope is equivalent to its slack matrix having minimum rank equal to dimension plus one and can be decided by an SDP relaxation of the corresponding rank-minimization problem.
Rothvoß, The matching polytope has exponential extension complexity , Journal of the ACM, 64 (2017), pp
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Determining inscribability of polytopes via rank minimization based on slack matrices
Inscribability of a polytope is equivalent to its slack matrix having minimum rank equal to dimension plus one and can be decided by an SDP relaxation of the corresponding rank-minimization problem.