  We study vantage-point trees constructed using an independent sample from the
uniform distribution on a fixed convex body $K$ in $(\mathbb{R}^d,\|\cdot\|)$,
where $\|\cdot\|$ is an arbitrary homogeneous norm on $\mathbb{R}^d$. We prove
that a sequence of sets, associated with the left boundary of a vantage-point
tree, forms a recurrent Harris chain on the space of convex bodies in
$(\mathbb{R}^d,\|\cdot\|)$. The limiting object is a ball polyhedron, that is,
an a.s.~finite intersection of closed balls in $(\mathbb{R}^d,\|\cdot\|)$ of
possibly different radii. As a consequence, we derive a limit theorem for the
length of the leftmost path of a vantage-point tree.
