I have recently learned the polar factorization theorem for vector-valued functions due to Brenier. Namely, given a probability space $(X,\mu)$ and a bounded domain $\Omega\subset \mathbb{R}^n$ with the normalized Lebesgue measure (i.e. $\mathcal{L}^n(\Omega)=1$). Then there is a "polar factorization'' for a function $u\colon X\to \mathbb{R}^n$ such that $u(x)=\nabla \phi(s(x))$ (with very less regularity), where $s\colon X\to \Omega$ is measure-preserving, and $\phi\colon \Omega\to \mathbb{R}$ is convex.

Brenier, Yann, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), no. 4, 375-417.

The proof I know uses the theory of optimal transportation, for instance can be found in

An elementary proof of the polar factorization of vector-valued functions. (English summary) Arch. Rational Mech. Anal. 128 (1994), no. 4, 381-399.

Since I am not very familiar with this field, I wonder whether there is some other known proof for this result. In particular, I am also interested in possible extensions of this result, say if we add more restrictions on $u$, can we conclude in the decomposition we have better regularity for $\phi$, say $\phi$ will be strictly convex, instead of convex? Similar for $s$.

I am very grateful for possible comments and suggestions.

  • 1
    $\begingroup$ Actually, the proof in 1994 reference does NOT use any optimal transportation theory, rather it is based on classical results in convexity theory. $\endgroup$ – Piyush Grover Dec 28 '15 at 16:30

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