  We introduce a new class of distances between nonnegative Radon measures in
Euclidean spaces. They are modeled on the dynamical characterization of the
Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou-Brenier and
provide a wide family interpolating between the Wasserstein and the homogeneous
(dual) Sobolev distances.
  From the point of view of optimal transport theory, these distances minimize
a dynamical cost to move a given initial distribution of mass to a final
configuration. An important difference with the classical setting in mass
transport theory is that the cost not only depends on the velocity of the
moving particles but also on the densities of the intermediate configurations
with respect to a given reference measure.
  We study the topological and geometric properties of these new distances,
comparing them with the notion of weak convergence of measures and the well
established Kantorovich-Rubinstein-Wasserstein theory. An example of possible
applications to the geometric theory of gradient flows is also given.
