On the Role of Cylindrical Functions in Kantorovich duality

TL;DR

This paper investigates the use of cylindrical functions in the dual formulation of the Monge-Kantorovich optimal transportation problem, especially in infinite-dimensional spaces, and explores conditions for their applicability with practical examples.

Contribution

It provides new insights into when cylindrical functions can be employed in Kantorovich duality in infinite-dimensional settings, extending the theoretical framework.

Findings

Conditions identified for using cylindrical functions in dual formulations

Examples demonstrating applications in infinite-dimensional spaces

Enhanced understanding of duality in optimal transportation

Abstract

We study the dual formulation of the Monge-Kantorovich optimal transportation problem, in particular under what circumstances it is permitted in an infinite dimensional setting to use cylindrical functions, i.e. functions of the form $\varphi\circ P$ where $P$ is a finite-rank operator and $\varphi$ is a smooth, compactly supported function. In the last section, some examples of applications are presented.