We provide sufficient conditions under which the center-outward distribution and quantile functions introduced in Chernozhukov et al. (2017) and Hallin (2017) are homeomorphisms, thereby extending a recent result by Figalli (2018). Our approach relies on Caffarelli’s classical regularity theory for the solutions of the Monge–Ampere equation, but has to deal with difficulties related with the unboundedness at the origin of the density of the spherical uniform reference measure. Our conditions are satisfied by probabilities on Euclidean space with a general (bounded or unbounded) convex support which are not covered by Figalli. We also provide some additional results about center-outward distribution and quantile functions, including the fact that quantile sets exhibit a limiting form of the so-called “lighthouse convexity” property.

中文翻译：

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关于基于最优传输的中心向外分布和分位数函数的规律性的注记

我们提供了充分条件，在该条件下，Chernozhukov 等人引入了中心向外分布和分位数函数。(2017) 和 Hallin (2017) 是同胚，从而扩展了 Figalli (2018) 最近的结果。我们的方法依赖于 Caffarelli 的经典正则性理论来求解 Monge-Ampere 方程，但必须处理与球面均匀参考度量密度原点处的无界性相关的困难。我们的条件由欧几里得空间上的概率满足，该空间具有一般（有界或无界）凸支撑，而 Figalli 未涵盖。我们还提供了一些关于中心向外分布和分位数函数的额外结果，包括分位数集表现出所谓的“灯塔凸性”属性的限制形式这一事实。