Suppose that X1, X2, X3,
… are independent random points in Rd
with common density f, having compact support Ω with smooth boundary
∂Ω, with f[mid ]Ω continuous. Let
Rni, k denote the distance from
Xi to its kth
nearest neighbour amongst the first n points, and let
Mn, k =
maxi[les ]nRni, k.
Let θ denote the volume of the unit ball. Then as n → ∞,
formula here
If instead the points lie in a compact smooth d-dimensional Riemannian manifold K, then
nθMdn, k/
log n → (minKf)−1, almost surely.