Ph.D Thesis | |

Ph.D Student | Bartz Seadia |
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Subject | On Abstract Convex Antiderivatives in Analysis and Geometry |

Department | Department of Mathematics |

Supervisor | PROFESSOR EMERITUS Simeon Reich |

We introduce, study and apply settings in abstract convex analysis which give rise to families of abstract convex antiderivatives. Given partial data regarding a c-subdifferential, we consider the family of all c-convex c-antiderivatives that comply with the given data. We prove that this family is not empty and contains both its lower and upper envelopes. We prove duality relations for such families and duality formulae for the envelopes. Furthermore, we construct these optimal c-convex c-antiderivatives explicitly. It turns out that these settings and results offer a unifying language for phenomena in different fields of analysis, which have so far been considered to be quite apart. Furthermore, employing the above results and settings we extend the theories in these fields.

Among other particular discussions, we present three main applications. In our first main application we embed the discussion of Lipschitz and Holder functions on general metric spaces in the framework of c-convexity. Then we formulate and solve a constrained Lipschitz extension problem. Given a function on a metric space, a set of pairs of which the given function preserves distances and a set of given values of the function, we extend these values to a Lipschitz function on the whole space while preserving the prescribed distances. We prove explicit formulae for the minimal and maximal extensions of which the most particular case recovers the well-known extensions of McShane and Whitney. Our second main application is in the theory of optimal transport. We show how our optimal c-convex c-antiderivatives play the role of monopolistic constrained optimal repricing strategies. This is then formulated as a sharpening of a restriction theorem for optimal transport plans of Villani. Thus, we cast this additional economical interpretation and we construct explicitly the minimal and maximal prices. Our third application is of the theory of representation of monotone operators by convex functions. Previously, initializing steps have been taken towards the extension of this theory to the general setting of c-monotonicity and c-convexity. We take the extension of the theory some steps farther. We associate a family of C-convex C-antiderivatives, defined on a product set, with a given c-monotone mapping. It turns out that the well-known Fitzpatrick function is precisely the minimal C-convex C-antiderivative in this family. This sheds new light on the underlying nature of representative functions of c-monotone mappings even in the well-studied classical case and closes the gap between the Fitzpatrick function and the Fitzpatrick family.