  Amenability is a notion of facial exposedness for convex cones that is
stronger than being facially dual complete (or "nice") which is, in turn,
stronger than merely being facially exposed. Hyperbolicity cones are a family
of algebraically structured closed convex cones that contain all spectrahedra
(linear sections of positive semidefinite cones) as special cases. It is known
that all spectrahedra are amenable. We establish that all hyperbolicity cones
are amenable. As part of the argument, we show that any face of a hyperbolicity
cone is a hyperbolicity cone. As a corollary, we show that the intersection of
two hyperbolicity cones, not necessarily sharing a common relative interior
point, is a hyperbolicity cone.
