Title | The Monge Problem for Distance Cost in Geodesic Spaces |

Publication Type | Journal Article |

Year of Publication | 2013 |

Authors | Bianchini, S, Cavalletti, F |

Journal | Communications in Mathematical Physics |

Volume | 318 |

Pagination | 615–673 |

Date Published | Mar |

ISSN | 1432-0916 |

Abstract | We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dLis a geodesic Borel distance which makes (X, dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1-dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting dL-cyclical monotonicity is not sufficient for optimality. |

URL | https://doi.org/10.1007/s00220-013-1663-8 |

DOI | 10.1007/s00220-013-1663-8 |

## The Monge Problem for Distance Cost in Geodesic Spaces

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