## Convexity in the proof of Brenier’s Polar Factorization Theorem (in Topics in Optimal Transportation)

I'm Reading Cédric Villani's Topics in Optimal Transportation. In his proof of Brenier's Polar factorization theorem he writes: "If we exhibit $s$ such that $s # \lambda = \nu$ and $s = \nabla \varphi \circ h$ for some convex function $\varphi$..."

Why can we assume such a representation exists? Why can we assume $\varphi$ is convex?

The proof in question: p.120

-
I don't think you can assume that such a representation exists. The point of the proof is to show that such an $s$ exists. – Deane Yang May 3 2011 at 3:49