When is the linear image of a closed convex cone closed? We present very simple, and intuitive necessary conditions, which unify, and generalize seemingly disparate, classical sufficient conditions: polyhedrality of the cone, and “Slater” type conditions are necessary and sufficient, when the dual cone belongs to a class, that we call nice cones. Nice cones subsume all cones amenable to treatment by efficient optimization algorithms: for instance, polyhedral, semidefinite, and p-cones. provide similarly attractive conditions for an equivalent problem: the closedness of the sum of two closed convex cones
