Improved bounds for the expected number of $k$-sets

Abstract

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 $\Delta\subset S$ of size $d$ such that the hyperplane spanned by $\Delta$ 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 B\'{a}r\'{a}ny 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 B\'{a}r\'{a}ny 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.