Motivated by Boroczky and Schneider's characterization of set polarity, and Artstein-Avidan and Milman's characterization of the Legendre transform, we look for extensions of the notions of set duality and functional duality. The latter, functional duality, has generalizations in the form of cost-transforms, from the theory of optimal transport. We will show how such costs can also give generalized set dualities, and show that in fact all set dualities are induced by a cost. We will demonstrate the power of this theory by finding costs generating many well known classes of bodies (e.g. unconditional bodies, reciprocals and flowers). Based on a joint work with Shiri Artstein-Avidan and Katarzyna Wyczesany.
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