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When is the linear image of a closed convex cone closed? We present very simple and intuitive necessary conditions that (1) unify, and generalize seemingly disparate, classical sufficient conditions such as 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, anddoi:10.1287/moor.1060.0242 fatcat:wbw4v4trzvgszgxrcslxx2lsca