  We consider the geometry of the space of Borel measures endowed with a
distance that is defined by generalizing the dynamical formulation of the
Wasserstein distance to concave, nonlinear mobilities. We investigate the
energy landscape of internal, potential, and interaction energies. For the
internal energy, we give an explicit sufficient condition for geodesic
convexity which generalizes the condition of McCann. We take an eulerian
approach that does not require global information on the geodesics. As
by-product, we obtain existence, stability, and contraction results for the
semigroup obtained by solving the homogeneous Neumann boundary value problem
for a nonlinear diffusion equation in a convex bounded domain. For the
potential energy and the interaction energy, we present a non-rigorous argument
indicating that they are not displacement semiconvex.
