  "Spatial" or "geometric" quantiles are the only multivariate quantiles coping
with both high-dimensional data and functional data, also in the framework of
multiple-output quantile regression. This work studies spatial quantiles in the
finite-dimensional case, where the spatial quantile $\mu_{\alpha,u}(P)$ of the
distribution $P$ taking values in $\mathbb{R}^d $ is a point in $\mathbb{R}^d$
indexed by an order $\alpha\in[0,1)$ and a direction $u$ in the unit sphere
$\mathcal{S}^{d-1}$ of $\mathbb{R}^d$ --- or equivalently by a vector $\alpha
u$ in the open unit ball of $\mathbb{R}^d$. Recently, Girard and Stupfler
(2017) proved that (i) the extreme quantiles $\mu_{\alpha,u}(P)$ obtained as
$\alpha\to 1$ exit all compact sets of $\mathbb{R}^d$ and that (ii) they do so
in a direction converging to $u$. These results help understanding the nature
of these quantiles: the first result is particularly striking as it holds even
if $P$ has a bounded support, whereas the second one clarifies the delicate
dependence of spatial quantiles on $u$. However, they were established under
assumptions imposing that $P$ is non-atomic, so that it is unclear whether they
hold for empirical probability measures. We improve on this by proving these
results under much milder conditions, allowing for the sample case. This
prevents using gradient condition arguments, which makes the proofs very
challenging. We also weaken the well-known sufficient condition for uniqueness
of finite-dimensional spatial quantiles.
