Covering random points in a unit disk

Abstract

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 .