Regularity of center-outward distribution functions in non-convex domains

For a probability in its center outward distribution function , introduced in V. Chernozhukov, A. Galichon, M. Hallin, and M. Henry (“Monge–Kantorovich depth, quantiles, ranks and signs,” , vol. 45, no. 1, pp. 223–256, 2017) and M. Hallin, E. del Barrio, J. Cuesta-Albertos, and C. Matrán (“Distribution and quantile functions, ranks and signs in dimension d: a measure transportation approach,” , vol. 49, no. 2, pp. 1139–1165, 2021), is a new and successful concept of multivariate distribution function based on mass transportation theory. This work proves, for a probability 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 and the continuity of its inverse, the quantile, . This relaxes the convexity assumption in E. del Barrio, A. González-Sanz, and M. Hallin (“A note on the regularity of optimal-transport-based center-outward distribution and quantile functions,” , vol. 180, p. 104671, 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.