Optimal transportation between hypersurfaces bounding some strictly convex domains

Abstract

Let $M,N$ be two smooth compact hypersurfaces of $\mathbb{R}^n$ which bound strictly convex domains equipped with two absolutely continuous measures $\mu$ and $\nu$ (with respect to the volume measures of $M$ and $N$). We consider the optimal transportation from $\mu$ to $\nu$ for the quadratic cost. Let $(\phi:m \to \mathbb{R},\psi:N \to \mathbb{R})$ be some functions which achieve the supremum in the Kantorovich formulation of the problem and which satisfy $$ \psi (y) = \inf_{z\in M} \Bigl( \frac{1}{2}|y-z|^2 -\varphi(z)\Bigr); \varphi (x)=\inf_{z\in N} \Bigl( \frac{1}{2}|x-z|^2 -\psi(z)\Bigr).$$ Define for $y \in N$, $$\varphi^\Box(y) = \sup_{z\in M} \Bigl( \frac{1}{2}|y-z|^2 -\varphi(z)\Bigr).$$ In this short paper, we exhibit a relationship between the regularity of $\varphi^\Box$ and the existence of a solution to the Monge problem.