I explain the formulation of the Wasserstein-Fisher-Rao distance, introduced a few years ago as the natural extension of the Wasserstein L2 metric to the space of positive Radon measures. We present the equivalence between the dynamic formulation and a static formulation, where we relax the marginal constraints with relative entropy. Then, we will present a connection with standard optimal transport on a cone space. The second part of the talk is motivated by this optimal transport distance and we study a generalized Camassa-Holm equation, for which we study existence of minimizing generalized geodesics, a la Brenier.