## Duke Mathematical Journal

### The Monge problem in ${\mathbb R}^d$

#### Abstract

We first 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. In the final part of the paper we show how to extend this existence result to a general open subset of ${\mathbb R}^d$.

#### Article information

Source
Duke Math. J. Volume 157, Number 3 (2011), 551-572.

Dates
First available in Project Euclid: 1 April 2011

https://projecteuclid.org/euclid.dmj/1301678733

Digital Object Identifier
doi:10.1215/00127094-1272939

Mathematical Reviews number (MathSciNet)
MR2785830

#### Citation

Champion, Thierry; De Pascale, Luigi. The Monge problem in ${\mathbb R}^d$. Duke Math. J. 157 (2011), no. 3, 551--572. doi:10.1215/00127094-1272939. https://projecteuclid.org/euclid.dmj/1301678733

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