Orthogonal projectional exposedness of homogeneous cones under a suitable inner product

Determine whether every homogeneous cone can be made orthogonally projectionally exposed by an appropriate change of inner product, and establish whether the symmetric cones are the only homogeneous cones with this property.

Background

The paper establishes that every homogeneous cone is projectionally exposed, but the projections constructed from the T-algebra framework need not be self-adjoint with respect to the T-algebra inner product. In contrast, symmetric cones are known to be orthogonally projectionally exposed under an appropriate inner product.

The unresolved issue is whether changing the inner product can always produce orthogonal projectional exposedness for an arbitrary homogeneous cone, and whether the property characterizes symmetric cones among homogeneous cones.