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

"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.