On the connection of facially exposed, and nice cones

A closed convex cone K is called nice, if the set K∗ + F⊥ is closed for all F faces of K, where K∗ is the dual cone of K, and F⊥ is the orthogonal complement of the linear span of F. The niceness property plays a role in the facial reduction algorithm of Borwein and Wolkowicz, and the question whether the linear image of a nice cone is closed also has a simple answer. We prove several characterizations of nice cones and show a strong connection with facial exposedness. We prove that a nice cone must be facially exposed; in reverse, facial exposedness with an added technical condition implies niceness. We conjecture that nice, and facially exposed cones are actually the same, and give some supporting evidence.