SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4783-4805, January 2020.
It is well known that the optimal transport problem on the real line for the classical distance cost may not have a unique solution. In this paper we recover uniqueness by considering the transport problems where the costs are a power smaller than one of the distance and letting this parameter tend to one. A complete construction of this solution that we call excursion coupling is given. This is reminiscent of the one in the convex case. It is also characterized as the solution of secondary transport problems. Moreover, a combinatoric/geometric characterization of the routes used for this transport plan is provided.

中文翻译：

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严凹案例对实线蒙格运输问题的解决

SIAM数学分析杂志，第52卷，第5期，第4783-4805页，2020年1月。
众所周知，对于经典距离成本而言，实线上的最优运输问题可能没有唯一的解决方案。在本文中，我们通过考虑运输成本小于距离之一的幂的运输问题并使该参数趋向于一来恢复唯一性。给出了该解决方案的完整构造，我们称之为偏移耦合。这使人联想到凸形外壳。它也可以解决二次运输问题。此外，提供了用于该运输计划的路线的组合/几何特征。