On the connection of facially exposed and nice cones

A closed convex cone KK in a finite dimensional Euclidean space is called nice   if the set K∗+F⊥K∗+F⊥ is closed for all FF faces of KK, where K∗K∗ is the dual cone of KK, and F⊥F⊥ is the orthogonal complement of the linear span of FF. The niceness property plays a role in the facial reduction algorithm of Borwein and Wolkowicz, and the question of whether the linear image of the dual 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; conversely, facial exposedness with an added condition implies niceness.We conjecture that nice, and facially exposed cones are actually the same, and give supporting evidence.