  Based on the novel concept of multivariate center-outward quantiles
introduced recently in Chernozhukov et al. (2017) and Hallin et al. (2021), we
are considering the problem of nonparametric multiple-output quantile
regression. Our approach defines nested conditional center-outward quantile
regression contours and regions with given conditional probability content
irrespective of the underlying distribution; their graphs constitute nested
center-outward quantile regression tubes. Empirical counterparts of these
concepts are constructed, yielding interpretable empirical regions and contours
which are shown to consistently reconstruct their population versions in the
Pompeiu-Hausdorff topology. Our method is entirely non-parametric and performs
well in simulations including heteroskedasticity and nonlinear trends; its
power as a data-analytic tool is illustrated on some real datasets.
