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