Set Reconstruction by Voronoi cells

Abstract

For a Borel set $A$ and a homogeneous Poisson point process $\eta$ in ${\mathbb{R}}^d$ of intensity $\lambda >0$, define the Poisson--Voronoi approximation $A_{\eta }$ of $A$ as a union of all Voronoi cells with nuclei from $\eta$ lying in $A$. If $A$ has a finite volume and perimeter we find an exact asymptotic of $\E\Vol(AΔA_η)$ as $\lambda \to \infty$ where $\Vol$ is the Lebesgue measure. Estimates for all moments of $\Vol(A_η)$ and $\Vol(AΔA_η)$ together with their asymptotics for large $\lambda$ are obtained as well.

19 pages, minor revisions