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Boundary density and Voronoi set estimation for irregular sets

In this paper, we study the inner and outer boundary densities of some sets with self-similar boundary having Minkowski dimension $s\textgreater{}d-1$ in $\mathbb{R}^{d}$. These quantities turn out to be crucial in some problems of set estimation theory, as we show here for the Voronoi approximation of the set with a random input constituted by $n$ iid points in some larger bounded domain. We prove that some classes of such sets have positive inner and outer boundary density, and therefore satisfy Berry-Essen bounds in $n^{-s/2d}$ for Kolmogorov distance. The Von Koch flake serves as an example, and a set with Cantor boundary as a counter-example. We also give the almost sure rate of convergence of Hausdorff distance between the set and its approximation.