Many empty triangles have a common edge

TL;DR

This paper investigates the maximum number of empty triangles sharing a common edge in a finite point set, establishing a lower bound on this degree for random points in a convex body.

Contribution

It proves that for random points in a convex body, the maximum degree of a pair of points in terms of empty triangles is at least proportional to n divided by log n.

Findings

Expected maximum degree is at least proportional to n/ln n.

Results hold for random samples from any fixed convex body.

Provides a probabilistic bound on empty triangle configurations.

Abstract

Given a finite point set $X$ in the plane, the degree of a pair $\{x,y\} \subset X$ is the number of empty triangles $t=conv\{x,y,z\}$, where empty means $t\cap X=\{x,y,z\}$. Define $deg X$ as the maximal degree of a pair in $X$. Our main result is that if $X$ is a random sample of $n$ independent and uniform points from a fixed convex body, then $deg X \ge cn/\ln n$ in expectation.