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 ${\mu }_0$ to another one ${\mu }_1$ consists of two parts: the first one measures the transportation from ${\mu }_0$ to an intermediate (pivot) measure $\mu$ to be determined (and subject to various constraints), and the second one measures the transportation from $\mu$ to ${\mu }_1$. This leads to Monge-Kantorovich interpolation problems under constraints for which we establish various properties of the optimal pivot measures $\mu$. 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.