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

This paper proves some results concerning the polar factorisation of an integrable vector-valued function $u$ into the composition $u = u^{\#} \circ s$, where $u^{\#} = \nabla \psi$ almost everywhere for some convex function $\psi$, and $s$ is 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 $u(x) \in \partial \psi (y)$ 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.