  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.
