We provide sufficient conditions under wich the center-outward distribution and quantile func-
tions introduced in Chernozhukov et al. (2017) and Hallin (2017) are homeomorphisms, thereby
extending a recent result by Figalli [12]. Our approach relies on Cafarelli’s classical regularity
theory for the solutions of the Monge-Amp`ere equation, but has to deal with difficulties related
with the unboundedness at the origin of the density of the spherical uniform reference measure.
Our conditions are satisfied by probabillities on Euclidean space with a general (bounded or un-
bounded) convex support which are not covered in [12]. We provide some additional results about
center-outward distribution and quantile functions, including the fact that quantile sets exhibit
some weak form of convexity.
