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For a Borel set A and a homogeneous Poisson point process η in 
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.
