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：对于$\mathbb{R}^d$上的概率测度$P$，其中心-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变量分布函数的新颖且成功的概念。本文证明：若概率测度$P$在其支撑集上密度局部有界且远离零和无穷，则中心-外映射在其支撑集内部连续，且其逆映射（即分位数$Q_{\pm}$）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-外映射稳定性刻画弱收敛性。