Covering random points in a unit disk

Jennie C. Hansen, Eric Schmutz, Li Sheng
2008 Advances in Applied Probability  
Let D be the punctured unit disk. It is easy to see that no pair x, y in D can cover D in the sense that D cannot be contained in the union of the unit disks centred at x and y. With this fact in mind, let V n = {X 1, X 2, ..., X n }, where X 1, X 2, ... are random points sampled independently from a uniform distribution on D. We prove that, with asymptotic probability 1, there exist two points in V n that cover all of V n .
doi:10.1017/s0001867800002354 fatcat:buklw6cmafbi5h4mm5nuq3hgzm