Poisson-Delaunay approximation

For a Borel set $A$ and a stationary Poisson point process ${\eta }_t$ in ${\mathbb{R}}^d$ of intensity $t>0$, the Poisson-Delaunay approximation $A_{{\eta }_t}$ of $A$ is the union of all Delaunay cells generated by ${\eta }_t$ with center in $A$. It is shown that ${\lambda }_d(A_{{\eta }_t})$ is an unbiased estimator for ${\lambda }_d(A)$, variance bounds and a quantitative central limit theorem are given. The asymptotic behaviour of the symmetric difference ${\lambda }_d(A\mathrm{\Delta }{A}_{{\eta }_t})$ is derived as $t\to \mathrm{\infty }$.