Spectrahedral Shadows

Abstract

In this chapter we introduce the notion of a spectrahedral shadow, and examine these sets thoroughly. They are still feasible sets for semidefinite programming, and in contrast to spectrahedra, many of the standard constructions from convexity theory can be applied to them. We explain how general semialgebraic sets can be approximated by spectrahedral shadows and how these methods provide criteria for being a spectrahedral shadow for many sets. While all convex semialgebraic sets in the plane are spectrahedral shadows, we also provide examples of convex semialgebraic sets that cannot be realized as a projection of a spectrahedron.