Large convex holes in random point sets
József Balogh, Hernàn González-Aguilar, Gelasio Salazar · arXiv (Cornell University) · 2012
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 $Θ(\log{n}/(\log{\log{n}}))$, regardless of the shape of $R$.