Projectional exposedness of spectrahedral and hyperbolicity cones

Determine whether all spectrahedral cones and all hyperbolicity cones are projectionally exposed.

Background

Homogeneous cones form a proper subclass of both spectrahedral cones and hyperbolicity cones. The paper proves projectional exposedness for homogeneous cones, while noting that the strongest general results currently available for the larger classes establish amenability rather than projectional exposedness.

The authors explicitly leave unresolved whether projectional exposedness extends to either of these two broader classes of convex cones.

References

Homogeneous cones are spectrahedral and, therefore, are hyperbolicity cones as well. These are two classes of cones that strictly contain homogeneous cones. Are they projectionally exposed?

— Faces of homogeneous cones and applications to homogeneous chordality  (2501.09581 - Gouveia et al., 16 Jan 2025) in Section 4, Conclusion and open questions, item (c)