On the Continuity of Center-Outward Distribution and Quantile Functions

To generalize the notion of distribution function to dimension $d\geq 2$, in the recent papers it was proposed a concept of center-outward distribution function based on optimal transportation ideas, and the inferential properties of the corresponding center-outward quantile function were studied. A crucial tool needed to derive the desired inferential properties is the continuity and invertibility for the center-outward quantile function outside the origin, as this ensures the existence of closed and nested quantile contours. The aim of this paper is to prove such a continuity and invertibility result.