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Spectrahedra are affine sections of the cone of positive semidefinitematrices which form a rich class of convex bodies that properly contains thatof polyhedra. While the class of polyhedra is closed under linear projections,the class of spectrahedra is not. In this paper we investigate the problem ofdeciding containment of projections of polyhedra and spectrahedra based onprevious works on containment of spectrahedra. The main concern is to studythese containment problems by formulating them as polynomial nonnegativityproblems. This allows to state hierarchies of (sufficient) semidefiniteconditions by applying (and proving) sophisticated Positivstellens\"atze. Wealso extend results on a solitary sufficient condition for containment ofspectrahedra coming from the polyhedral situation as well as connections to thetheory of (completely) positive linear maps.

Containment Problems for Projections of Polyhedra and Spectrahedra