\] into $P^{v}_{\text{in}}$ and $P^{v}_{\text{out}}$ where $\textbf{v}$ is the parent node. Store only the single point $\text{rep}_v = m \in P$ in node $\textbf{v}$, the center of the ball $B(m,r_0)$. We continue this partitioning until we have nodes with only a single point contained in them.  Since each child contains at least $\frac{n}{c}$ points, each subset reduces by a factor of at least $0 - \frac{0}{c}$ at each step, and hence the depth of the tree is logarithmic. We calculate the depth more exactly, noting that in Lemma \[