Реконструкция множеств с помощью мозаики Вороного

(joint work with Matthias Reitzner and Dmitry Zaporozhets)
Let $A$ be a bounded Borel set in $\mathbb{R}^d$ with finite perimeter which can not be observed directly. Let $\Phi$ be a homogeneous Poisson point process in $\mathbb{R}^d$ of observation points with intensity $\lambda>0$. The only information about $A$ at our disposal is which points of $\Phi$ lie within $A$, and which do not. The Poisson Voronoi reconstruction of $A$ is the union of Voronoi cells of all points of $\Phi$ within $A$. We consider the quality of this reconstruction by measuring the volume $v_\lambda$ of the symmetric set difference between the set itself and its reconstructed version. We prove the exact asymptotic of the mean of $v_\lambda$, and an upper bound for the asymptotic of its variance as $\lambda\to\infty$.