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>d-1 in \mathbb{R}^{d}. These quantities turn out to be crucial in some problems of set estimation, 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-Esseen bounds in n^{-s/2d} for Kolmogorov distance. The Von Koch flake serves as an example, and a set with Cantor boundary as a counterexample. We also give the almost sure rate of convergence of Hausdorff distance between the set and its approximation.