Set Reconstruction by Voronoi cells

Details Set Reconstruction by Voronoi cells Set Reconstruction by Voronoi cells

2011-11-17

arxiv.org/abs/1111.4169v2

Mathematics

Probability

19 pages, minor revisions

Scientific paper

For a Borel set $A$ and a homogeneous Poisson point process $\eta$ in $\R^d$ 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 $\E\Vol(A\Delta A_\eta)$ as $\lambda\to\infty$ where $\Vol$ is the Lebesgue measure. Estimates for all moments of $\Vol(A_\eta)$ and $\Vol(A\Delta A_\eta)$ together with their asymptotics for large $\lambda$ are obtained as well.

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