Containment Problems for Projections of Polyhedra and Spectrahedra

Spectrahedra are affine sections of the cone of positive semidefinite matrices which form a rich class of convex bodies that properly contains that of polyhedra. While the class of polyhedra is closed under linear projections, the class of spectrahedra is not. In this paper we investigate the problem of deciding containment of projections of polyhedra and spectrahedra based on previous works on containment of spectrahedra. The main concern is to study these containment problems by formulating them as polynomial nonnegativity problems. This allows to state hierarchies of (sufficient) semidefinite conditions by applying (and proving) sophisticated Positivstellens\"atze. We also extend results on a solitary sufficient condition for containment of spectrahedra coming from the polyhedral situation as well as connections to the theory of (completely) positive linear maps.