Random Euclidean coverage from within.

Let X 1 , X 2 , ... be independent random uniform points in a bounded domain A ⊂ R d with smooth boundary. Define the coverage threshold R n to be the smallest r such that A is covered by the balls of radius r centred on X 1 , ... , X n . We obtain the limiting distribution of R n and also a strong law of large numbers for R n in the large-n limit. For example, if A has volume 1 and perimeter | ∂ A | , if d = 3 then P [ n π R n 3 - log n - 2 log (log n) ≤ x ] converges to exp (- 2 - 4 π 5 / 3 | ∂ A | e - 2 x / 3) and (n π R n 3) / (log n) → 1 almost surely, and if d = 2 then P [ n π R n 2 - log n - log (log n) ≤ x ] converges to exp (- e - x - | ∂ A | π - 1 / 2 e - x / 2) . We give similar results for general d, and also for the case where A is a polytope. We also generalize to allow for multiple coverage. The analysis relies on classical results by Hall and by Janson, along with a careful treatment of boundary effects. For the strong laws of large numbers, we can relax the requirement that the underlying density on A be uniform. [ABSTRACT FROM AUTHOR]