Optimal transportation for a quadratic cost with convex constraints and applications

2011/10/14 by Filippo Santambrogio, Jimenez, Chloé, Santambrogio, Filippo
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1110.3237

openalex publication_date 2011/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove existence of an optimal transport map in the Monge-Kantorovich problem associated to a cost c(x,y) which is not finite everywhere, but coincides with |x-y|² if the displacement y-x belongs to a given convex set C and it is +∞ otherwise. The result is proven for C satisfying some technical assumptions allowing any convex body in \R² and any convex polyhedron in \R^d, d>2. The tools are inspired by the recent Champion-DePascale-Juutinen technique. Their idea, based on density points and avoiding disintegrations and dual formulations, allowed to deal with L^∞ problems and, later on, with the Monge problem for arbitrary norms.

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