The positive semidefinite (psd) rank of a
polytope is the size of the smallest psd cone that admits an affine slice that projects linearly onto the
polytope. The psd rank of a
d -
polytope is at least
d+1, and when equality holds we say that the
polytope is psd-minimal. In this paper we develop new tools for the study of psd-minimality and use them to give a complete classification of psd-minimal 4-
polytopes. The main tools introduced are trinomial obstructions, a new algebraic obstruction for psd-minimality, and the slack ideal of a
polytope, which encodes the space of realizations of a
polytope up to projective equivalence.
Our central result is that there are 31 combinatorial classes of psd-minimal 4-polytopes. We provide combinatorial information and an explicit psd-minimal realization in each class. For 11 of these classes, every polytope in them is psd-minimal, and these are precisely the combinatorial classes of the known projectively unique 4-polytopes. We give a complete characterization of psd-minimality in the remaining classes, encountering in the process counterexamples to some open conjectures.