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 $$t=\mathrm {conv} \{x,y,z\},$$ 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.