A spectrahedron is the intersection of the cone of positive semidefinite matrices with a linear space. The computational problem of maximizing a linear functional over a spectrahedron is known as semidefinite programming. If the linear space consists of diagonal matrices only, then we recover the familiar concepts of convex polyhedra and linear programming. In this lecture, we explore the combinatorial geometry of spectrahedra and related convex bodies. Topics include multifocal ellipses and characterizations of spectrahedra and their projections due to Helton, Nie and Vinnikov. We also discuss work with Sanyal and Sottile on orbitopes, that is, convex hulls of orbits of compact real algebraic groups.
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