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An application of the universality theorem for Tverberg partitions to data depth and hitting convex sets
Computational Geometry ( IF 0.4 ) Pub Date : 2020-03-16 , DOI: 10.1016/j.comgeo.2020.101649
Imre Bárány , Nabil H. Mustafa

We show that, as a consequence of a new result of Pór on universal Tverberg partitions, any large-enough set P of points in Rd has a (d+2)-sized subset whose Radon point has half-space depth at least cd|P|, where cd(0,1) depends only on d. We then give two applications of this result. The first is to computing weak ϵ-nets by random sampling. The second is to show that given any set P of points in Rd and a parameter ϵ>0, there exists a set of O(ϵd2+1) d2-dimensional simplices (ignoring polylogarithmic factors) spanned by points of P such that they form a transversal for all convex objects containing at least ϵ|P| points of P.



中文翻译:

Tverberg分区的普遍性定理在数据深度和命中凸集上的应用

我们证明,由于Pór在通用Tverberg分区上的新结果,任何足够大的P点集[Rd 有一个 d+2子集的Radon点至少具有半空间深度 Cd|P|,在哪里 Cd01个仅取决于d。然后我们给出该结果的两个应用。首先是通过随机抽样计算弱ϵ网络。第二个是表明给定任何点P[Rd 和一个参数 ϵ>0,存在一组 Øϵ-d2+1个 d2P点跨越的三维单形(忽略多对数因子),使得它们对所有至少包含的凸对象形成一个横向ϵ|P|P点。

更新日期:2020-03-16
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