  The centerpoint theorem is a well-known and widely used result in discrete
geometry. It states that for any point set $P$ of $n$ points in $\mathbb{R}^d$,
there is a point $c$, not necessarily from $P$, such that each halfspace
containing $c$ contains at least $\frac{n}{d+1}$ points of $P$. Such a point
$c$ is called a centerpoint, and it can be viewed as a generalization of a
median to higher dimensions. In other words, a centerpoint can be interpreted
as a good representative for the point set $P$. But what if we allow more than
one representative? For example in one-dimensional data sets, often certain
quantiles are chosen as representatives instead of the median. We present a
possible extension of the concept of quantiles to higher dimensions. The idea
is to find a set $Q$ of (few) points such that every halfspace that contains
one point of $Q$ contains a large fraction of the points of $P$ and every
halfspace that contains more of $Q$ contains an even larger fraction of $P$.
This setting is comparable to the well-studied concepts of weak
$\varepsilon$-nets and weak $\varepsilon$-approximations, where it is stronger
than the former but weaker than the latter.
