Monge–Kantorovich interpolation with constraints and application to a parking problem

We consider optimal transport problems where the cost for transporting a given probability measure μ₀ to another one ( μ₁ ) consists of two parts: the first one measures the transportation from μ₀ to an intermediate (pivot) measure μ to be determined (and subject to various constraints), and the second one measures the transportation from μ to μ₁ . This leads to Monge–Kantorovich interpolation problems under constraints for which we establish various properties of the optimal pivot measures μ . Considering the more general situation where only some part of the mass uses the intermediate stop leads to a mathematical model for the optimal location of a parking region around a city. Numerical simulations, based on entropic regularization, are presented both for the optimal parking regions and for Monge–Kantorovich constrained interpolation problems.