Smooth hyperbolicity cones are second-order cone representable

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 our second theorem, according to which every compact convex semialgebraic set with Nash-smooth boundary of strict positive curvature is socr. The proof uses the technique of tensor evaluation.