Semiconvexity of (weak) Kantorovich potentials in the Lorentzian optimal transport problem

2025/11/07 by Alec Metsch, Metsch, Alec
Mathematics · #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #math.DG #math.OC

paper · pdf · doi:10.48550/arxiv.2511.05227

openalex publication_date 2025/11/07 · openalex created_date 2025/11/11 · openalex updated_date 2026/07/28

Abstract

We study semiconvexity properties of (weak) Kantorovich potentials for the Lorentzian optimal transport problem with the standard cost function c. We show that, in general, this regularity - known in the Riemannian context - does not extend to the Lorentzian setting. Nevertheless, we provide a general regularity result for c-convex functions and show that, under suitable general assumptions on the measures, this yields semiconvexity of the (weak) potentials at least on an open set of full measure. This, in turn, allows us to conclude the existence and uniqueness of an optimal transport map.

Citations

• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• Ricci curvature for metric-measure spaces via optimal transport
• The geometry of optimal transportation
• Polar factorization and monotone rearrangement of vector‐valued functions

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• On the Uniqueness of Kantorovich Potentials
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