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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.