Proximity complexes and filtrations are central constructions in topological data analysis. Built using distance functions, or more generally metrics, they are often used to infer connectivity information from point clouds. Here we investigate proximity complexes and filtrations built over the Chebyshev metric, also known as the maximum metric or ${\ell }_{\infty }$ metric, rather than the classical Euclidean metric. Somewhat surprisingly, the ${\ell }_{\infty }$ case has not yet been investigated thoroughly. In this paper, we examine a number of classical complexes under this metric, including the Čech, Vietoris-Rips, and Alpha complexes. We define two new families of flag complexes, which we call the Alpha flag and Minibox complexes, and prove their equivalence to Čech complexes in homological degrees zero and one. Moreover, we provide algorithms for finding Minibox edges of two, three, and higher-dimensional points. Finally, we present computational experiments on random points, which shows that Minibox filtrations can often be used to speed up persistent homology computations in homological degrees zero and one by reducing the number of simplices in the filtration.

邻近复合体和过滤是拓扑数据分析的核心结构。使用距离函数或更一般的度量构建，它们通常用于从点云推断连接信息。在这里，我们研究建立在 Chebyshev 度量上的邻近复合体和过滤，也称为最大度量或${\ell }_{\infty }$度量，而不是经典的欧几里得度量。让人有些意外的是，${\ell }_{\infty }$案件尚未彻底调查。在本文中，我们研究了该度量下的许多经典配合物，包括 Čech、Vietoris-Rips 和 Alpha 配合物。我们定义了两个新的标志复合体系列，我们称之为 Alpha 标志和 Minibox 复合体，并证明它们与 Čech 复合体在 0 和 1 的同调度上是等价的。此外，我们提供了用于查找二、三和更高维点的 Minibox 边的算法。最后，我们展示了随机点的计算实验，这表明 Minibox 过滤通常可用于通过减少过滤中的单纯形数量来加速 0 和 1 同源度中的持久同源性计算。