Let $P$ be a set of $n$ points in $\Re^2$. For a parameter $\varepsilon\in (0,1)$, a subset $C\subseteq P$ is an \emph{$\varepsilon$-kernel} of $P$ if the projection of the convex hull of $C$ approximates that of $P$ within $(1-\varepsilon)$-factor in every direction. The set $C$ is a \emph{weak $\varepsilon$-kernel} of $P$ if its directional width approximates that of $P$ in every direction. Let $\mathsf{k}_{\varepsilon}(P)$ (resp.\ $\mathsf{k}^{\mathsf{w}}_{\varepsilon}(P)$) denote the minimum-size of an $\varepsilon$-kernel (resp. weak $\varepsilon$-kernel) of $P$. We present an $O(n\mathsf{k}_{\varepsilon}(P)\log n)$-time algorithm for computing an $\varepsilon$-kernel of $P$ of size $\mathsf{k}_{\varepsilon}(P)$, and an $O(n^2\log n)$-time algorithm for computing a weak $\varepsilon$-kernel of $P$ of size ${\mathsf{k}}^{\mathsf{w}}_{\varepsilon}(P)$. We also present a fast algorithm for the Hausdorff variant of this problem. In addition, we introduce the notion of \emph{$\varepsilon$-core}, a convex polygon lying inside $\mathsf{ch}(P)$, prove that it is a good approximation of the optimal $\varepsilon$-kernel, present an efficient algorithm for computing it, and use it to compute an $\varepsilon$-kernel of small size.

翻译：设$P$为$\Re^2$中$n$个点的集合。对于参数$\varepsilon\in (0,1)$，子集$C\subseteq P$是$P$的$\varepsilon$-核，若$C$的凸包投影在任意方向上以$(1-\varepsilon)$因子逼近$P$的凸包投影。子集$C$是$P$的弱$\varepsilon$-核，若其方向宽度在任意方向上逼近$P$的方向宽度。记$\mathsf{k}_{\varepsilon}(P)$（分别地$\mathsf{k}^{\mathsf{w}}_{\varepsilon}(P)$）为$P$的$\varepsilon$-核（分别地弱$\varepsilon$-核）的最小基数。我们提出计算$P$的基数为$\mathsf{k}_{\varepsilon}(P)$的$\varepsilon$-核的$O(n\mathsf{k}_{\varepsilon}(P)\log n)$时间算法，以及计算基数为${\mathsf{k}}^{\mathsf{w}}_{\varepsilon}(P)$的弱$\varepsilon$-核的$O(n^2\log n)$时间算法。此外，我们针对该问题的Hausdorff变体提出快速算法。同时，我们引入$\varepsilon$-核的概念——位于$\mathsf{ch}(P)$内部的凸多边形，证明其是最优$\varepsilon$-核的良好近似，给出高效计算算法，并用于计算小尺寸的$\varepsilon$-核。