Euler Sprays and Wasserstein Geometry of the Space of Shapes

We study a distance between shapes defined by minimizing the integral of kinetic energy along transport paths constrained to measures with characteristic-function densities. The formal geodesic equations for this shape distance are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The minimization problem exhibits an instability associated with microdroplet formation, with the following outcomes: Shape distance is equal to Wasserstein distance. Furthermore, any two shapes of equal volume can be approximately connected by an Euler spray—a countable superposition of ellipsoidal droplet solutions of incompressible Euler equations. Every Wasserstein geodesic between shape densities is a weak limit of Euler sprays. Each Wasserstein geodesic is also the unique minimizer of a relaxed least-action principle for a fluid-vacuum mixture.