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When is the linear image of a closed convex cone closed? Despite being fundamental in convex analysis, this problem has received suprisingly little attention. 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 todoi:10.1287/moor.1060.0242 fatcat:wbw4v4trzvgszgxrcslxx2lsca