  Quantiles are a fundamental concept in probability and theoretical statistics
and a daily tool in their applications. While the univariate concept of
quantiles is quite clear and well understood, its multivariate extension is
more problematic. After half a century of continued efforts and many proposals,
two concepts, essentially, are emerging: the so-called (relabeled) geometric
quantiles, extending the characterization of univariate quantiles as minimizers
of an L1 loss function involving the check functions, and the more recent
center-outward quantiles based on measure transportation ideas. These two
concepts yield distinct families of quantile regions and quantile contours. Our
objective here is to present a comparison of their main theoretical properties
and a numerical investigation of their differences.
