  The Helton-Nie Conjecture (HNC) is the proposition that every convex
semialgebraic set is a spectrahedral shadow. Here we prove that HNC is
equivalent to another propo- sition related to quadratically constrained
quadratic programming. Namely, that the convex hull of the rank-one elements of
any spectrahedron is a spectrahedral shadow. In the case of compact convex
semialgebraic sets, the spectrahedra may be taken to be compact. We illustrate
the relationship between spetrahedra and these convex subsets with examples.
