Rectifiability of Optimal Transportation Plans

Abstract

The purpose of this note is to show that the solution to the Kantorovich optimal transportation problem is supported on a Lipschitz manifold, provided the cost is $C^2$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.

References in corpus (3)

• Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory
• On strict convexity and $C^1$ regularity of potential functions in optimal transportation
• Extremal doubly stochastic measures and optimal transportation

Cited by in corpus (6)

• Sectional and intermediate Ricci curvature lower bounds via Optimal Transport
• Insights into capacity constrained optimal transport
• A glimpse into the differential topology and geometry of optimal transport
• On the local structure of optimal measures in the multi-marginal optimal transportation problem
• Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy
• Pseudo-Riemannian geometry calibrates optimal transportation