Let $\mathcal{P}$ be a simple polygon with $m$ vertices and let $P$ be a set of $n$ points inside $\mathcal{P}$. We prove that there exists, for any $\varepsilon>0$, a set $\mathcal{C} \subset P$ of size $O(1/\varepsilon^2)$ such that the following holds: for any query point $q$ inside the polygon $\mathcal{P}$, the geodesic distance from $q$ to its furthest neighbor in $\mathcal{C}$ is at least $1-\varepsilon$ times the geodesic distance to its further neighbor in $P$. Thus the set $\mathcal{C}$ can be used for answering $\varepsilon$-approximate furthest-neighbor queries with a data structure whose storage requirement is independent of the size of $P$. The coreset can be constructed in $O\left(\frac{1}{\varepsilon} \left( n\log(1/\varepsilon) + (n+m)\log(n+m)\right) \right)$ time.

翻译：设$\mathcal{P}$是一个包含$m$个顶点的简单多边形，$P$是$\mathcal{P}$内$n$个点的集合。我们证明，对于任意$\varepsilon>0$，存在一个大小为$O(1/\varepsilon^2)$的子集$\mathcal{C} \subset P$，使得以下性质成立：对于多边形$\mathcal{P}$内的任意查询点$q$，$q$到$\mathcal{C}$中最远邻点的测地距离至少是$q$到$P$中最远邻点测地距离的$1-\varepsilon$倍。因此，集合$\mathcal{C}$可用于回答$\varepsilon$-近似最远邻查询，且所需数据结构的存储需求与$P$的大小无关。该核心集可在$O\left(\frac{1}{\varepsilon} \left( n\log(1/\varepsilon) + (n+m)\log(n+m)\right) \right)$时间内构建完成。