Second-order cone representable slices of the positive semidefinite cone of size three
Abstract
To demonstrate the discrepancy between second-order cone and semidefinite programming, Hamza Fawzi showed that the cone S+3 of symmetric positive semidefinite matrices of size 3 is not second-order cone representable (socr). A slice of S+3 is intersection of S+3 and a linear sub-space of the space S3 of 3 × 3 symmetric matrices. It is known that some slices of S+3 are socr, while some others are not. We classify socr slices of S+3 by showing that a slice of S+3 is socr if and if it has dimension at most 4 or is orthogonal to a non-zero singular matrix (where the orthogonality is considered with respect to the standard trace scalar product).
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