Set reconstruction by Voronoi cells

Abstract

For a Borel set A and a homogeneous Poisson point process η in ∝^d of intensity λ>0, define the Poisson--Voronoi approximation A_η of A as a union of all Voronoi cells with nuclei from η lying in A. If A has a finite volume and perimeter, we find an exact asymptotic of E Vol(AΔ A_η) as λ→∞, where Vol is the Lebesgue measure. Estimates for all moments of Vol(A_η) and Vol(AΔ A_η) together with their asymptotics for large λ are obtained as well.