  We study the boundary structure of closed convex cones, with a focus on
facially dual complete (nice) cones. These cones form a proper subset of
facially exposed convex cones, and they behave well in the context of duality
theory for convex optimization. Using the well-known and commonly used concept
of tangent cones in nonlinear optimization, we introduce some new notions for
exposure of faces of convex sets. Based on these new notions, we obtain a
necessary condition and a sufficient condition for a cone to be facially dual
complete. In our sufficient condition, we utilize a new notion called
lexicographic tangent cones (these are a family of cones obtained from a
recursive application of the tangent cone concept). Lexicographic tangent cones
are related to Nesterov's lexicographic derivatives and to the notion of
subtransversality in the context of variational analysis.
