Multimarginal Optimal Transport Maps for One–dimensional Repulsive Costs
نویسندگان
چکیده
منابع مشابه
Partial Regularity for Optimal Transport Maps
We prove that, for general cost functions on R, or for the cost d/2 on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.
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The Monge-Kantorovich problem is to move one distribution of mass onto another as efficiently as possible, where Monge’s original criterion for efficiency [19] was to minimize the average distance transported. Subsequently studied by many authors, it was not until 1976 that Sudakov showed solutions to be realized in the original sense of Monge, i.e., as mappings from R to R [23]. A second proof...
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In the special case “cost=squared distance” on R, the problem was solved by Caffarelli [Caf1, Caf2, Caf3, Caf4], who proved the smoothness of the map under suitable assumptions on the regularity of the densities and on the geometry of their support. However, a major open problem in the theory was the question of regularity for more general cost functions, or for the case “cost=squared distance”...
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We prove some stability results concerning the smoothness of optimal transport maps with general cost functions. In particular, we show that the smoothness of optimal transport maps is an open condition with respect to the cost function and the densities. As a consequence, we obtain regularity for a large class of transport problems where the cost does not necessarily satisfy the MTW condition.
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In the first part of the paper we briefly decribe the classical problem, raised by Monge in 1781, of optimal transportation of mass. We discuss also Kantorovich's weak solution of the problem, which leads to general existence results, to a dual formulation, and to necessary and sufficient optimality conditions. In the second part we describe some recent progress on the problem of the existence ...
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ژورنال
عنوان ژورنال: Canadian Journal of Mathematics
سال: 2015
ISSN: 0008-414X,1496-4279
DOI: 10.4153/cjm-2014-011-x