Calculus of Variations and Geometric Measure Theory

T. Champion - L. De Pascale

The Monge problem in $R^d$

created by depascal on 05 Jun 2009
modified on 27 Jul 2010


Accepted Paper

Inserted: 5 jun 2009
Last Updated: 27 jul 2010

Journal: Duke Mathematical Journal
Year: 2009


We consider the Monge problem in a convex bounded subset of $\mathbb{R}^d$. The cost is given by a general norm, and we prove the existence of an optimal transport map under the classical assumption that the first marginal is absolutely continuous with respect to the Lebesgue measure. The approach we propose to solve this problem does not use the disintegration of measures.

Version revised on the October 16 2009 according to a first referee report.

Keywords: Monge-Kantorovich problem, optimal transport problem, cyclical monotonicity