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

TL;DR

This paper investigates bounds on the expected number of $k$-sets and $k$-facets in random point sets, providing new asymptotic bounds in the plane and exploring the tightness of existing techniques.

Contribution

It establishes new bounds for the expected number of $k$-facets in planar distributions and analyzes the tightness of Bárány and Steiger's technique for convex set systems.

Findings

Proves $E_P(k,n) = O(n(k+1)^{1/4})$ for planar distributions with measure-zero lines.

Shows the technique by Bárány and Steiger is tight for certain convex set systems.

Provides bounds for the expected number of $k$-sets in translated convex bodies, up to logarithmic factors.

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.