  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.
