$ \newcommand{\epsA}{\Mh{\delta}} \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \renewcommand{\P}{P} \newcommand{\diam}{\Delta} \newcommand{\Mh}[1]{#1} \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\polylog}{\mathrm{polylog}} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}[2]{\left\| {#1} - {#2} \right\|} \newcommand{\PP}{P} \newcommand{\ptq}{q} \newcommand{\pts}{s}$Given a set $P \subset \Re^d$ of $n$ points, with diameter $\diam$, and a parameter $\epsA \in (0,1)$, it is known that there is a partition of $P$ into sets $P_1, \ldots, P_t$, each of size $O(1/\epsA^2)$, such that their convex-hulls all intersect a common ball of radius $\epsA \diam$. We prove that a random partition, with a simple alteration step, yields the desired partition, resulting in a (randomized) linear time algorithm. We also provide a deterministic algorithm with running time $O( dn \log n)$. Previous proofs were either existential (i.e., at least exponential time), or required much bigger sets. In addition, the algorithm and its proof of correctness are significantly simpler than previous work, and the constants are slightly better. We also include a number of applications and extensions using the same central ideas. For example, we provide a linear time algorithm for computing a ``fuzzy'' centerpoint, and prove a no-dimensional weak $\eps$-net theorem with an improved constant.

翻译：给定一个包含 $n$ 个点、直径为 $\diam$ 的点集 $P \subset \Re^d$，以及参数 $\epsA \in (0,1)$，已知存在一个将 $P$ 划分为子集 $P_1, \ldots, P_t$（每个子集大小为 $O(1/\epsA^2)$）的划分，使得这些子集的凸包均与半径为 $\epsA \diam$ 的公共球相交。我们证明，通过引入一个简单的修正步骤，随机划分即可得到所需的划分，从而产生一种（随机化的）线性时间算法。我们还提供了一种运行时间为 $O( dn \log n)$ 的确定性算法。先前的证明要么是存在性的（即至少需要指数时间），要么需要大得多的子集。此外，本文的算法及其正确性证明比先前的工作显著简单，且常数略有优化。我们还基于相同核心思想给出了一系列应用和推广。例如，我们提供了一种计算“模糊”中心点的线性时间算法，并证明了具有改进常数的无维度弱 $\eps$-网定理。