$ \newcommand{\epsA}{\Mh{\delta}} \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \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 $\PP \subset \Re^d$ of $n$ points, with diameter $\diam$, and a parameter $\epsA \in (0,1)$, it is known that there is a partition of $\PP$ into sets $\PP_1, \ldots, \PP_t$, 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 linear time algorithm. 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. In addition, we provide a linear time algorithm for computing a ``fuzzy'' centerpoint. We also prove a no-dimensional weak $\eps$-net theorem with an improved constant.

翻译：给定一个由$n$个点组成的集合$\PP \subset \Re^d$，其直径为$\diam$，并给定参数$\epsA \in (0,1)$，已知存在一种将$\PP$划分为集合$\PP_1, \ldots, \PP_t$的方法，每个集合的大小为$O(1/\epsA^2)$，使得它们的凸包均与一个半径为$\epsA \diam$的公共球相交。我们证明，通过一个简单的修正步骤，随机划分即可得到所需的划分，从而产生一个线性时间算法。以往的证明要么是存在性的（即至少需要指数时间），要么需要大得多的集合。此外，该算法及其正确性证明比以往工作简单得多，且常数略有改进。同时，我们提供了一个计算“模糊”中心点的线性时间算法。我们还证明了一个常数改进后的无维度弱$\eps$-网定理。