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 $(\varphi :m\to \mathbb{R},\psi :N\to \mathbb{R})$ be some functions which achieve the supremum in the Kantorovich formulation of the problem and which satisfy $$\psi (y)=\underset{z\in M}{\inf }(\frac{1}{2}|y-z{|}^2-\phi (z));\phi (x)=\underset{z\in N}{\inf }(\frac{1}{2}|x-z{|}^2-\psi (z)).$$ Define for $y\in N$, $${\phi }^{\square }(y)=\underset{z\in M}{\sup }(\frac{1}{2}|y-z{|}^2-\phi (z)).$$ In this short paper, we exhibit a relationship between the regularity of ${\phi }^{\square }$ and the existence of a solution to the Monge problem.