Many empty triangles have a common edge

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 $degX$ 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 $degX\ge cn/\ln n$ in expectation.