The Monge problem in R^n, with a possibly asymmetric norm cost function and absolutely continuous first marginal, is generally underdetermined. An optimal transport plan is selected by a secondary variational problem, from a work on crystalline norms. In this way the mass still moves along lines. The paper provides a quantitative absolute continuity push forward estimate for the translation along these lines: the consequent area formula, for the disintegration of the Lebesgue measure w.r.t. the partition into these 1D-rays, shows that the conditional measures are absolutely continuous, and yields uniqueness of the optimal secondary transport plan non-decreasing along rays, recovering that it is induced by a map.
A proof of Monge problem in Rn by stability
CARAVENNA, LAURA
2011
Abstract
The Monge problem in R^n, with a possibly asymmetric norm cost function and absolutely continuous first marginal, is generally underdetermined. An optimal transport plan is selected by a secondary variational problem, from a work on crystalline norms. In this way the mass still moves along lines. The paper provides a quantitative absolute continuity push forward estimate for the translation along these lines: the consequent area formula, for the disintegration of the Lebesgue measure w.r.t. the partition into these 1D-rays, shows that the conditional measures are absolutely continuous, and yields uniqueness of the optimal secondary transport plan non-decreasing along rays, recovering that it is induced by a map.
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