On the existence of $L^p$-Optimal Transport maps for norms on $\mathbb{R}^N$

TL;DR

This paper establishes the existence of $L^p$-optimal transport maps for a class of branching norms on $R^N$, introducing cylinder-like convex functions and solving the Monge problem for specific cost functions.

Contribution

It introduces the notion of cylinder-like convex functions and proves existence of $L^p$-optimal transport maps for branching norms on $R^N$, including all norms in $R^2$ and crystalline norms.

Findings

Existence of $L^p$-optimal transport maps for branching norms.

Introduction of cylinder-like convex functions.

Applicability to all norms in $R^2$ and crystalline norms.

Abstract

In this paper, we prove existence of $L^p$-optimal transport maps with $p \in (1,\infty)$ in a class of branching metric spaces defined on $\mathbb{R}^N$. In particular, we introduce the notion of cylinder-like convex function and we prove an existence result for the Monge problem with cost functions of the type $c(x, y) = f(g(y - x))$, where $f: [0, \infty) \rightarrow [0, \infty)$ is an increasing strictly convex function and $g: \mathbb{R}^N \rightarrow [0, \infty)$ is a cylinder-like convex function. When specialised to cylinder-like norm, our results shows existence of $L^p$-optimal transport maps for several "branching'" norms, including all norms in $\mathbb{R}^2$ and all crystalline norms.