On the existence of L^p-Optimal Transport maps for norms on ℝ^N

2024/10/29 by Guoxi Liu, Liu, Guoxi, Mattia Magnabosco +3
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2410.22576

Abstract

In this paper, we prove existence of L^p-optimal transport maps with p ∈ (1,∞) in a class of branching metric spaces defined on 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, ∞) → [0, ∞) is an increasing strictly convex function and g: ℝ^N → [0, ∞) 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 ℝ² and all crystalline norms.

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