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

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 $f:[0,\infty )\to [0,\infty )$ is an increasing strictly convex function and $g:{\mathbb{R}}^N\to [0,\infty )$ 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.