An elementary proof of the polar factorization of vector-valued functions

Abstract

We present an elementary proof of an important result of Y. Brenier [Br1, Br2], namely, that vector fields in ℝd satisfying a nondegeneracy condition admit the polar factorization (*) u(x)=▽ψ(s(x)), where ψ is a convex function and s is a measure-preserving mapping. Brenier solves a minimization problem using Monge-Kantorovich theory; whereas we turn our attention to a dual problem, whose Euler-Lagrange equation turns out to be (*).