Geometric Lax Conjecture

A hyperbolicity region is the connected component containing the reference point of the set where a real homogeneous polynomial is nonzero. A spectrahedron is a set defined by a linear matrix inequality. Geometric Lax Conjecture. Every hyperbolicity region is a spectrahedron.

In full generality, the conjecture is open, although it holds in dimension two by the theorem of Helton and Vinnikov. A proof would classify spectrahedra through the behavior of their boundary surfaces.