  The use of quantiles to obtain insights about multivariate data is addressed.
It is argued that incisive insights can be obtained by considering directional
quantiles, the quantiles of projections. Directional quantile envelopes are
proposed as a way to condense this kind of information; it is demonstrated that
they are essentially halfspace (Tukey) depth levels sets, coinciding for
elliptic distributions (in particular multivariate normal) with density
contours. Relevant questions concerning their indexing, the possibility of the
reverse retrieval of directional quantile information, invariance with respect
to affine transformations, and approximation/asymptotic properties are studied.
It is argued that the analysis in terms of directional quantiles and their
envelopes offers a straightforward probabilistic interpretation and thus
conveys a concrete quantitative meaning; the directional definition can be
adapted to elaborate frameworks, like estimation of extreme quantiles and
directional quantile regression, the regression of depth contours on
covariates. The latter facilitates the construction of multivariate growth
charts---the question that motivated all the development.
