We show that, as a consequence of a new result of Pór on universal Tverberg partitions, any large-enough set P of points in ${\mathbb{R}}^d$ has a $(d+2)$-sized subset whose Radon point has half-space depth at least $c_d\cdot |P|$, where $c_d\in (0,1)$ 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 ${\mathbb{R}}^d$ and a parameter $ϵ>0$, there exists a set of $O({ϵ}^{-\lfloor \frac{d}{2}\rfloor }+1)$ $\lfloor \frac{d}{2}\rfloor$-dimensional simplices (ignoring polylogarithmic factors) spanned by points of P such that they form a transversal for all convex objects containing at least $ϵ\cdot |P|$ points of P.

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${\mathbb{[R}}^d$ 有一个 $（d+2）$子集的Radon点至少具有半空间深度 $C_d\cdot |P|$，在哪里 $C_d\in （0，\mathrm{1个}）$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${\mathbb{[R}}^d$ 和一个参数 $ϵ>0$，存在一组 $Ø（{ϵ}^{-\lfloor \frac{d}{2}\rfloor }+\mathrm{1个}）$ $\lfloor \frac{d}{2}\rfloor$P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$ϵ\cdot |P|$P点。