Calculus of Variations and Geometric Measure Theory
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L. Ambrosio - S. Rigot

Optimal Mass Transportation in the Heisenberg Group

created on 21 Nov 2002
modified on 17 Dec 2002


Accepted Paper

Inserted: 21 nov 2002
Last Updated: 17 dec 2002

Journal: Journal of Functional Analysis
Year: 2002


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.


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