We study the $k$-th nearest neighbor distance function from a finite point-set in $\mathbb{R}^d$. We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric characterization for critical points and their indices, along with detailed information about the possible changes in homology at the critical levels. We conclude by computing the expected number of critical points for a homogeneous Poisson process. Our results deliver significant insights and tools for the analysis of persistent homology in order-$k$ Delaunay mosaics, and random $k$-fold coverage.

翻译：我们研究$\mathbb{R}^d$中有限点集的第$k$个最近邻距离函数。我们提供了一个Morse理论框架来分析其子水平集的拓扑结构。具体而言，我们给出了临界点及其指数的简洁组合几何刻画，并详细描述了临界水平处同调群可能发生的变化。最后，我们计算了齐次泊松过程临界点数量的期望值。我们的结果为分析$k$阶Delaunay镶嵌的持续同调以及随机$k$重覆盖提供了重要见解和工具。