No Selection Lemma for Empty Triangles

TL;DR

This paper investigates empty triangles in point sets, disproving a conjecture by showing that the maximum number of empty triangles incident to any point aligns with known upper bounds, even in the case of empty triangles.

Contribution

The paper proves that the upper bound on the number of empty triangles incident to any point applies to empty triangles, countering Bárány's conjecture.

Findings

Existence of point sets with many empty triangles where no point is in more than O(n^{3-2α}) empty triangles.

Disproof of Bárány's conjecture regarding the incidence of empty triangles.

Confirmation that the upper bound for empty triangles matches the general case.

Abstract

Let $S$ be a set of $n$ points in general position in the plane. The Second Selection Lemma states that for any family of $\Theta(n^3)$ triangles spanned by $S$, there exists a point of the plane that lies in a constant fraction of them. For families of $\Theta(n^{3-\alpha})$ triangles, with $0\le \alpha \le 1$, there might not be a point in more than $\Theta(n^{3-2\alpha})$ of those triangles. An empty triangle of $S$ is a triangle spanned by $S$ not containing any point of $S$ in its interior. B\'ar\'any conjectured that there exist an edge spanned by $S$ that is incident to a super constant number of empty triangles of $S$. The number of empty triangles of $S$ might be $O(n^2)$; in such a case, on average, every edge spanned by $S$ is incident to a constant number of empty triangles. The conjecture of B\'ar\'any suggests that for the class of empty triangles the above upper bound might not hold. In this paper we show that, somewhat surprisingly, the above upper bound does in fact hold for empty triangles. Specifically, we show that for any integer $n$ and real number $0\leq \alpha \leq 1$ there exists a point set of size $n$ with $\Theta(n^{3-\alpha})$ empty triangles such that any point of the plane is only in $O(n^{3-2\alpha})$ empty triangles.