Covering random points in a unit disk
Jennie C. Hansen, Eric Schmutz, Sheng Li · Advances in Applied Probability · 2008
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 .