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.
Steiner, Systematische Entwicklung der Abh¨ angigkeit geometrischer Gestalten von einander, Fincke, 1832
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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.