We present an extension of the Wasserstein L2 distance to the space of positive Radon measures as an infimal convolution between the Wasserstein L2 metric and the Fisher-Rao metric. In the work of Brenier, optimal transport has been developed in its study of the incompressible Euler equation. For the Wasserstein-Fisher-Rao metric, the corresponding fluid dynamic equation is known as the Camassa-Holm equation (at least in dimension 1), originally introduced as a geodesic flow on the group of diffeomorphisms. This point of view provides an isometric embedding of the group of diffeomorphisms endowed with this right-invariant metric in the automorphisms group of the fiber bundle of half densities endowed with an L 2 type of cone metric. As a direct consequence, we describe a new polar factorization on the automorphism group of half-densities which can be seen as a constrained version of Brenier’s theorem. The main application consists in writing the Camassa-Holm equation on S^1 as a particular case of the incompressible Euler equation on a group of homeomorphisms of R^2 that preserve a radial density which has a singularity at 0, the cone point.
