When is the linear image of a closed convex cone closed? We present very simple, and intuitive necessary conditions, which 1) unify, and generalize seemingly disparate, classical sufficient conditions: polyhedrality of the cone, and ``Slater'' type conditions; 2) 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. 3) provide similarly attractive conditions for an equivalent problem: the closedness of the sum of two closed convex cones.

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