Smooth hyperbolicity cones are second-order cone representable

Abstract: Netzer and Sanyal proved that every smooth hyperbolicity cone is a spectrahedral shadow. We generalize and sharpen this result at the same time, by showing that every Nash-smooth hyperbolicity cone is even second-order cone representable (socr). The result is proved as a consequence of another theorem, according to which every compact convex semialgebraic set is socr, provided that its boundary is Nash-smooth of strict positive curvature. The proof uses the technique of tensor evaluation.