Non-existence of polar factorisations and polar inclusion of a vector-valued mapping

TL;DR

This paper investigates the conditions under which vector-valued functions can be polar factored or included via measure-preserving mappings, revealing limitations and extending the theoretical framework in optimal transport.

Contribution

It demonstrates that not all integrable functions admit polar factorizations, introduces the concept of polar inclusion, and characterizes measure-preserving plans as minimizers of a Monge-Kantorovich problem.

Findings

Not all integrable functions have polar factorizations.

Introduces the concept of polar inclusion for vector-valued functions.

Characterizes measure-preserving plans as solutions to an optimal transport problem.

Abstract

This paper proves some results concerning the polar factorisation of an integrable vector-valued function u into the composition of the gradient of a convex function with a measure-preserving mapping. Not every integrable function has a polar factorisation; we extend the class of counterexamples. We introduce a generalisation: u has a polar inclusion if u(x) belongs to the subdifferential of the convex function at y for almost every pair (x,y) with respect to a measure-preserving plan. Given a regularity assumption, we show that such measure-preserving plans are exactly the minimisers of a Monge-Kantorovich optimisation problem.