For a probability P in $R^d$ its center outward distribution function $F_{\pm}$, introduced in Chernozhukov et al. (2017) and Hallin et al. (2021), is a new and successful concept of multivariate distribution function based on mass transportation theory. This work proves, for a probability P with density locally bounded away from zero and infinity in its support, the continuity of the center-outward map on the interior of the support of P and the continuity of its inverse, the quantile, $Q_{\pm}$. This relaxes the convexity assumption in del Barrio et al. (2020). Some important consequences of this continuity are Glivenko-Cantelli type theorems and characterisation of weak convergence by the stability of the center-outward map.

翻译：设概率测度P定义在$R^d$上，其中心向外分布函数$F_{\pm}$由Chernozhukov等(2017)和Hallin等(2021)提出，是基于质量传输理论的多元分布函数新概念。本文证明：当概率测度P的密度在其支撑集内局部有界远离零和无穷时，中心向外映射在P支撑集内部连续，且其逆映射——分位数函数$Q_{\pm}$也连续。这一结果放宽了del Barrio等(2020)中的凸性假设。该连续性的重要推论包括Glivenko-Cantelli型定理以及通过中心向外映射稳定性刻画弱收敛性。