Set-valued recursions arising from vantage-point trees

Abstract

We study vantage-point trees constructed using an independent sample from the uniform distribution on a fixed convex body $K$ in $({\mathbb{R}}^d,\Vert \cdot \Vert )$, where $\Vert \cdot \Vert$ is an arbitrary 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 $({\mathbb{R}}^d,\Vert \cdot \Vert )$. The limiting object is a ball polyhedron, that is, an a.s.~finite intersection of closed balls in $({\mathbb{R}}^d,\Vert \cdot \Vert )$ of possibly different radii. As a consequence, we derive a limit theorem for the length of the leftmost path of a vantage-point tree.

15 pages