Helton–Nie Conjecture

A convex semialgebraic set is a convex subset of Euclidean space describable by polynomial equalities and inequalities. A spectrahedral shadow is a linear image of a spectrahedron. Helton–Nie Conjecture. Every convex semialgebraic set is a spectrahedral shadow.

Spectrahedral shadows are convex and semialgebraic and are closed under many natural operations, but the conjecture would establish that these properties characterize them. The supplied context does not state whether the conjecture has been resolved.