Large convex holes in random point sets

TL;DR

This paper investigates the expected size of the largest convex hole in a set of randomly chosen points within a convex region, revealing it grows logarithmically with the number of points.

Contribution

It establishes a precise asymptotic behavior for the expected number of vertices of the largest convex hole in random point sets, independent of the region's shape.

Findings

Expected size of largest convex hole is Θ(log n / log log n)

Result holds for any convex region R

Expected number of vertices grows slowly with n

Abstract

A {\em convex hole} (or {\em empty convex polygon)} of a point set $P$ in the plane is a convex polygon with vertices in $P$, containing no points of $P$ in its interior. Let $R$ be a bounded convex region in the plane. We show that the expected number of vertices of the largest convex hole of a set of $n$ random points chosen independently and uniformly over $R$ is $\Theta(\log{n}/(\log{\log{n}}))$, regardless of the shape of $R$.