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 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.

翻译：我们研究在固定凸体$K$（位于赋范空间$(\mathbb{R}^d,\|\cdot\|)$中，其中$\|\cdot\|$是$\mathbb{R}^d$上的任意范数）上，利用均匀分布独立样本构建的视点树。我们证明，与视点树左边界相关联的一系列集合，在$(\mathbb{R}^d,\|\cdot\|)$中凸体构成的空间上形成一个递归哈里斯链。其极限对象是一个球多面体，即$(\mathbb{R}^d,\|\cdot\|)$中不同半径闭球的几乎必然有限交集。由此，我们推导出视点树最左路径长度的极限定理。