Empty non-convex and convex four-gons in random point sets

Abstract

LetSbe a set ofnpoints distributed uniformly and independently in a convex, bounded set in the plane. A four-gon is called empty if it contains no points ofSin its interior. We show that the expected number of empty non-convex four-gons with vertices fromSis 12n2logn+o(n2logn) and the expected number of empty convex four-gons with vertices fromSis Θ(n2).