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This paper proves some results concerning the polar factorisation of anintegrable vector-valued function u into the composition of the gradient of aconvex function with a measure-preserving mapping. Not every integrablefunction has a polar factorisation; we extend the class of counterexamples. Weintroduce a generalisation: u has a polar inclusion if u(x) belongs to thesubdifferential of the convex function at y for almost every pair (x,y) withrespect to a measure-preserving plan. Given a regularity assumption, we showthat such measure-preserving plans are exactly the minimisers of aMonge-Kantorovich optimisation problem.

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