In this paper, we first define ray increasing and decreasing monotonicity of maps. If 𝑇 is an optimal transport map for the Monge problem with cost function ${\parallel y-x\parallel }_{\mathrm{sc}}$ in $R^n$ or 𝑇 is an optimal transport map for the Monge problem with cost function $d(x,y)$, the geodesic distance, in more general, non-branching geodesic spaces 𝑋, we show respectively equivalence of some previously introduced monotonicity properties and the property of ray increasing as well as ray decreasing monotonicity which we define in this paper. Then, by solving secondary variational problems associated with strictly convex and concave functions respectively, we show that there exist ray increasing and decreasing optimal transport maps for the Monge problem with cost function ${\parallel y-x\parallel }_{\mathrm{sc}}$. Finally, we give the classification of optimal transport maps for the Monge problem such that the cost function ${\parallel y-x\parallel }_{\mathrm{sc}}$ further satisfies the uniform smoothness and convexity estimates. That is, all of the optimal transport maps for such Monge problem can be divided into three different classes: the ray increasing map, the ray decreasing map and others.

中文翻译：

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射线单调性的等价性和严格凸范数的最优输运图的分类

在本文中，我们首先定义了光线增加和减少地图的单调性。如果𝑇是具有成本函数的蒙格问题的最佳运输图${\parallel ÿ--X\parallel }_{\mathrm{SC}}$ 在 ${\mathrm{[R}}^{ñ}$ 或𝑇是具有成本函数的蒙格问题的最佳运输图 $d（X，ÿ）$测地距离，在更一般的非分支测地空间𝑋中，我们分别显示了本文中定义的一些先前引入的单调性的等价性以及射线增加与减少的单调性的等价性。然后，通过分别求解分别与严格的凸函数和凹函数有关的二次变分问题，我们表明存在具有成本函数的蒙格问题的射线增加和减少的最优输运图${\parallel ÿ--X\parallel }_{\mathrm{SC}}$。最后，我们给出了蒙格问题的最优运输图的分类，使得成本函数${\parallel ÿ--X\parallel }_{\mathrm{SC}}$进一步满足均匀平滑度和凸度估计。也就是说，针对这种蒙格问题的所有最佳传输图可以分为三类：射线增加图，射线减少图和其他。