Inserted: 21 nov 2002
Last Updated: 17 dec 2002
Journal: Journal of Functional Analysis
In this paper we show existence and uniqueness of the optimal transport map in Monge--Kantorovich problem in the case when the ambient space is the Heisenberg group and the cost function is the square of a distance. In fact, we consider the two cases of the Carnot--Carathéodory distance and the Korányi distance. In the former case, by a careful analysis of the minimizing geodesics and the derivatives of the squared disance function, we show that our map is the limit of McCann's maps relative to the canonical approximating Riemannian manifolds. The same analysis also shows that McCann's representation formula still holds, with a suitable ``metric'' exponential map.