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"Spatial" or "geometric" quantiles are the only multivariate quantiles copingwith both high-dimensional data and functional data, also in the framework ofmultiple-output quantile regression. This work studies spatial quantiles in thefinite-dimensional case, where the spatial quantile $\mu_{\alpha,u}(P)$ of thedistribution $P$ 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 $\alphau$ 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 soin a direction converging to $u$. These results help understanding the natureof these quantiles: the first result is particularly striking as it holds evenif $P$ has a bounded support, whereas the second one clarifies the delicatedependence of spatial quantiles on $u$. However, they were established underassumptions imposing that $P$ is non-atomic, so that it is unclear whether theyhold for empirical probability measures. We improve on this by proving theseresults under much milder conditions, allowing for the sample case. Thisprevents using gradient condition arguments, which makes the proofs verychallenging. We also weaken the well-known sufficient condition for uniquenessof finite-dimensional spatial quantiles.

On the behavior of extreme $d$-dimensional spatial quantiles under minimal assumptions