A closed convex cone K is called nice, if the set K^* + F^\perp is closed for
all F faces of K, where K^* is the dual cone of K, and F^\perp is the
orthogonal complement of the linear span of F. The niceness property is
important for two reasons: it 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 condition implies niceness.
We conjecture that nice, and facially exposed cones are actually the same,
and give supporting evidence
