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In order to classify partial entanglement of multi-partite states, it isnatural to consider the convex hulls, intersections and differences of basicconvex cones obtained from partially separable states with respect topartitions of systems. In this paper, we consider convex cones consisting ofX-shaped three qubit states arising in this way. The class of X-shaped statesincludes important classes like Greenberger-Horne-Zeilinger diagonal states. Wefind all the extreme rays of those convex cones to exhibit correspondingpartially separable states. We also give characterizations for those coneswhich give rise to necessary criteria in terms of diagonal and anti-diagonalentries for general three qubit states.

On the convex cones arising from classifications of partial entanglement in the three qubit system