Georgia Institute of Technology College of Sciences

In the classical theory of optimal transport, Legendre duality arises naturally, as seen for example in Kantorovich’s duality theorem. Extending this idea to a general cost function naturally leads to a broader notion of functional cost-duality and the associated class of c-functions.

Similarly, in the setting of sets, taking polars provides an analogous notion of duality, mapping to the class of convex sets. In this talk, I will introduce cost dualities for sets and show that they correspond precisely to all order-reversing involutions on sets. Finally, I will explore the connections between c-duality and various geometric and functional inequalities.