  The aim of this paper is twofold. On one hand the generalized Minkowski sets
are defined and characterized. On the other hand, the Motzkin decomposable
sets, along with their epigraphic versions are considered and characterized in
new ways. Among them, the closed convex sets with one single minimal face, i.e.
translated closed convex cones, along with their epigraphic counterparts are
particularly studied.
