On the convexity theory of generating functions

TL;DR

This paper extends convexity theory for generating functions in optimal transportation, providing a geometric approach and analyzing regularity conditions, with implications for near field geometric optics.

Contribution

It introduces a geometric treatment of convexity theory for general generating functions, expanding the framework of optimal transportation and analyzing regularity conditions.

Findings

Extended convexity theory to more general generating functions.

Provided a geometric approach alternative to differential inequalities.

Analyzed invariance of regularity conditions under duality.

Abstract

In this paper, we extend our convexity theory for $C^2$ cost functions in optimal transportation to more general generating functions, which were originally introduced by the second author to extend the framework of optimal transportation to embrace near field geometric optics. In particular we provide an alternative geometric treatment to the previous analytic approach using differential inequalities, which also gives a different derivation of the invariance of the fundamental regularity conditions under duality. We also extend our local theory to cover the strict version of these conditions for $C^2$ cost and generating functions.