On semidefinite representations of non-closed sets

Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinitely representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. There are several recent results on the question which sets are semidefinitely representable. So far, all results focus on the case of closed sets.In this work we develop a new method to prove semidefinite representability of sets which are not closed. For example, the interior of a semidefinitely representable set is shown to be semidefinitely representable. More general, one can remove faces of a semidefinitely representable set and preserve semidefinite representability, as long as the faces are parametrized in a suitable way.