  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.
