Dualization

In the realm of convexity, almost every mathematical object can be paired with another, said to be dual to it. The pairing between convex cones and their polars has already been fundamental in the variational geometry of Chapter 6 in relating tangent vectors to normal vectors. The pairing between convex sets and sublinear functions in Chapter 8 has served as the vehicle for expressing connections between subgradients and subderivatives. Both correspondences are rooted in a deeper principle of duality for ‘conjugate’ pairs of convex functions, which will emerge fully here.

On the basis of this duality, close connections between otherwise disparate properties are revealed. It will be seen for instance that the level boundedness of one function in a conjugate pair corresponds to the finiteness of the other function around the origin. A catalog of such surprising linkages can be put together, and lists of dual operations and constructions to go with them.