Calculus of Variations and Geometric Measure Theory

A. Figalli

On the Continuity of Center-Outward Distribution and Quantile Functions

created by figalli on 15 May 2018


Accepted Paper

Inserted: 15 may 2018
Last Updated: 15 may 2018

Journal: Nonlinear Anal.
Year: 2018


To generalize the notion of distribution function to dimension $d\geq 2$, in some recent papers it has been proposed a concept of center-outward distribution function based on optimal transportation ideas, and the inferential properties of the corresponding center-outward quantile function have been 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.