*Preprint*

**Inserted:** 18 jan 2009

**Last Updated:** 19 jan 2009

**Year:** 2009

**Notes:**

Preprint SISSA 64*2008*M

**Abstract:**

We establish a solution to the Monge problem in ${R}^{N}$, with an asymmetric, {strictly convex} norm cost function, when the initial measure is absolutely continuous. We focus on the strategy, based on disintegration of measures, initially proposed by Sudakov. As known, there is a gap to fill. The missing step is completed when the unit ball is strictly convex, but not necessarily differentiable nor uniformly convex. The key disintegration is achieved following a similar proof for a variational problem.

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