  Representations of nonnegative polynomials as sums of squares are central to
real algebraic geometry and the subject of active research. The sum-of-squares
representations of a given polynomial are parametrized by the convex body of
positive semidefinite Gram matrices, called the Gram spectrahedron. This is a
fundamental object in polynomial optimization and convex algebraic geometry. We
summarize results on sums of squares that fit naturally into the context of
Gram spectrahedra, present some new results, and highlight related open
questions. We discuss sum-of-squares representations of minimal length and
relate them to Hermitian Gram spectrahedra and point evaluations on toric
varieties.
