I'm taking kind of a crash-course in convex analysis to complement my mathematical skills and was wondering if anyone knew about nice ways in which this kind of tools were used in Economics. To be more precise, some of the things I've seen so far are not strictly in the area of convex analysis but are very related, like dual spaces, weak topology, subdiferentials and Hahn-Banach theorem.
The only example I know of is the duality between the UMP and EMP in consumer theory (and of course the firm maximization and cost minimization problems). I also think that Hahn-Banach is used in the proof of the first welfare theorem.
Has anyone here used this kind of mathematical concepts in their work or has seen any interesting recent usage of them?