On the Closedness of the Linear Image of a Closed Convex Cone

• 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.