Orthogonal projectional exposedness of homogeneous cones under a changed inner product

Determine whether every homogeneous cone admits an inner product under which it is orthogonally projectionally exposed, and characterize whether symmetric cones are the only homogeneous cones with this property.

Background

The paper establishes that every homogeneous cone is projectionally exposed, meaning that each face is the image of the cone under an idempotent linear projection. However, the projections constructed from the T-algebra framework need not be self-adjoint with respect to the T-algebra inner product.

Symmetric cones are known to be orthogonally projectionally exposed under a suitable inner product, whereas the general case is unresolved. The open problem therefore asks both whether an appropriate change of inner product always suffices and whether that property characterizes symmetric cones among homogeneous cones.

References

We do not know whether an arbitrary homogeneous cone can become orthogonally projectionally exposed by changing the inner product appropriately. Currently, the only homogeneous cones known to be orthogonally projectionally exposed are symmetric cones Proposition~33, are those the only ones?

— 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 (b)