Optimal transportation between hypersurfaces bounding some strictly convex domains

TL;DR

This paper investigates the regularity properties of solutions to the optimal transportation problem between hypersurfaces bounding convex domains, linking the smoothness of a specific potential function to the existence of Monge solutions.

Contribution

It establishes a relationship between the regularity of a potential function and the existence of optimal transport maps between convex hypersurfaces.

Findings

Regularity of al^ox influences the existence of Monge solutions.

The paper characterizes conditions under which optimal transport maps exist for convex hypersurfaces.

Provides insights into the structure of optimal transportation potentials in convex geometric settings.

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.