On the existence of L^p-optimal transport maps for norms on 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 \(f\colon [0, \infty) \to [0, \infty)\) is an increasing strictly convex function and \(g\colon \mathbb{R}^N \to [0, \infty)\) 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.