We propose the convex floating body membership problem, which consists of efficiently determining when a query point $a\in\mathbb{R}^d$ belongs to the so-called $\varepsilon$-convex floating body of a given bounded convex domain $K\subset\mathbb{R}^d$. We consider this problem in an approximate setting, i.e., given a parameter $\delta>0$, the query can be answered either way if the Hilbert distance in $K$ of $a$ from the boundary of a relatively-scaled $\varepsilon$-convex floating body is less than $\delta$. We present a data structure for this problem that has storage size $O(\delta^{-d}\varepsilon^{-(d-1)/2})$ and achieves query time of $O({\delta^{-1}}\ln 1/\varepsilon)$. Our construction is motivated by a recent work of Abdelkader and Mount on APM queries, and relies on a comparison of convex floating bodies with balls in the Hilbert metric on $K$.

翻译：本文提出凸浮动体成员查询问题，该问题旨在高效判定查询点$a\in\mathbb{R}^d$是否属于给定有界凸域$K\subset\mathbb{R}^d$的所谓$\varepsilon$-凸浮动体。我们在近似设定下研究该问题：给定参数$\delta>0$，当$a$与相对缩放$\varepsilon$-凸浮动体边界的希尔伯特距离小于$\delta$时，查询可接受任意判定结果。我们提出一种数据结构，其存储复杂度为$O(\delta^{-d}\varepsilon^{-(d-1)/2})$，查询时间复杂度为$O({\delta^{-1}}\ln 1/\varepsilon)$。该构造受Abdelkader和Mount近期关于APM查询研究的启发，并基于凸浮动体与$K$上希尔伯特度量中球体的比较分析。