Equivalence between some definitions for the optimal mass transport problem and for the transport density on manifolds

In this paper we consider three problems, which are related to the classical Monges optimal mass transport problem and which are known to be equivalent when the ambient space is an open, convex and bounded subset of ⁿ; to these problems there correspond different definitions of particular measures (often called transport densities), which are also known to be equivalent. Here we will generalize the setting of these problems and the resulting definitions of transport densities to the case of a Riemannian manifold endowed with a finslerian semidistance, and we will see that the equivalences still hold. Mathematics Subject Classification (2000) 46G10, 49Q20, 58C35