  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 spectrahedral
cones (linear sections of positive semidefinite cones) as special cases. It is
known that all spectrahedral cones 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.
