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 thresholdR _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[nπRn3-logn-2log(logn)≤x] converges to exp (- 2 ^{- 4}π ^{5 / 3}| ∂A| e ^{- 2 x / 3}) and (nπRn3)/(logn)→1 almost surely, and if d= 2 then P[nπRn2-logn-log(logn)≤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.