  This work is concerned with different aspects of spectrahedra and their
projections, sets that are important in semidefinite optimization. We prove
results on the limitations of so called Lasserre and theta body relaxation
methods for semialgebraic sets and varieties. As a special case we obtain the
main result of the paper "Exposed faces of semidefinite representable sets" of
Netzer, Plaumann and Schweighofer. We also solve the open problems from that
work. We further prove some helpful facts which can not be found in the
existing literature, for example that the closure of a projection of a
spectrahedron is again such a projection. We give a unified account of several
results on convex hulls of curves and images of polynomial maps. We finally
prove a Positivstellensatz for projections of spectrahedra, which exceeds the
known results that only work for basic closed semialgebraic sets.
