$ \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：给定点集 $P \subset \Re^d$ 包含 $n$ 个点，直径为 $\diam$，以及参数 $\epsA \in (0,1)$，已知存在一种将 $P$ 划分为子集 $P_1, \ldots, P_t$ 的方法，每个子集的大小为 $O(1/\epsA^2)$，使得它们的凸包均与半径为 $\epsA \diam$ 的某个公共球相交。我们证明，通过一种简单的调整步骤，随机划分即可得到所需划分，从而产生（随机化）线性时间算法。我们还提供了一种确定性算法，其运行时间为 $O( dn \log n)$。先前的证明要么是存在性的（即至少指数时间复杂度），要么需要更大的子集。此外，该算法及其正确性证明比以往工作显著更简单，且常数略优。我们还基于相同核心思想给出了若干应用与扩展。例如，我们提供了一种计算“模糊”中心点的线性时间算法，并证明了一个具有改进常数的无维数弱 $\eps$-网定理。