Generalized Convexity and Transportation Theory
This chapter first presents the generalization of convexity theory when replacing duality products with general coupling functions on arbitrary sets. The notions of Fenchel conjugates, cyclical monotonicity and duality of optimization problems, have a natural extension to this setting, in which the augmented Lagrangian approach has a natural interpretation. Convex functions over measure spaces, constructed as Fenchel conjugates of integral functions of continuous functions, are shown to be sometimes equal to some integral of a function of their density. This is used in the presentation of optimal transportation theory over compact sets, and the associated penalized problems. The chapter ends with a discussion of the multi-transport setting.