$k$ -集期望个数的改进界
Improved bounds for the expected number of $k$-sets
摘要：
给定一个有限的点集$S\subset\mathbb{R}^d$，$S$的$k$ -集是一个大小为$k$的子集$A \subset S$，它可以通过一个超平面与$S \setminus A $严格分离。类似地，点集$S$在一般位置上的$k$ -刻面是大小为$d$的子集$Δ\subset S$，使得由$Δ$跨越的超平面在一侧具有来自$S$的$k$点。对于概率分布$P$在$\mathbb{R}^d$上，我们研究了$E_P(k,n)$，$k$随机点样本的期望$k$ -刻面数。当$P$是$\mathbb{R}^2$上的一个分布，使得每条线的测度为0时，我们证明了$E_P(k,n) = O(n(k+1)^{1/4})$。我们的论证是基于Bárány和Steiger的一种技术。我们研究它是如何可能的，以提高这一界限使用连续版本的多项式分割定理。这激发了一个问题有关的交点的代数曲线和$k$ -边图的一组点。我们还研究了$k$ -集问题的一个变化的集合系统的范围组成的所有平移的一些严格凸体在平面上。其动机是表明，该技术由Bárány和Steiger是紧的一个自然家庭的设置系统。对于任何这样的一组系统，我们确定的预期数量的$k$ -集是紧对数因子的界限。
英文摘要：
Given a finite set of points $S\subset\mathbb{R}^d$, a $k$-set of $S$ is a subset $A \subset S$ of size $k$ which can be strictly separated from $S \setminus A $ by a hyperplane. Similarly, a $k$-facet of a point set $S$ in general position is a subset $Δ\subset S$ of size $d$ such that the hyperplane spanned by $Δ$ has $k$ points from $S$ on one side. For a probability distribution $P$ on $\mathbb{R}^d$, we study $E_P(k,n)$, the expected number of $k$-facets of a sample of $n$ random points from $P$. When $P$ is a distribution on $\mathbb{R}^2$ such that the measure of every line is 0, we show that $E_P(k,n) = O(n(k+1)^{1/4})$. Our argument is based on a technique by Bárány and Steiger. We study how it may be possible to improve this bound using the continuous version of the polynomial partitioning theorem. This motivates a question concerning the points of intersection of an algebraic curve and the $k$-edge graph of a set of points. We also study a variation on the $k$-set problem for the set system whose set of ranges consists of all translations of some strictly convex body in the plane. The motivation is to show that the technique by Bárány and Steiger is tight for a natural family of set systems. For any such set system, we determine bounds for the expected number of $k$-sets which are tight up to logarithmic factors.
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