In this paper, we study the inner and outer boundary densities of some sets with self-similar boundary having Minkowski dimension
s
>
d
−
1
s>d-1
in
R
d
\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
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
/
2
d
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.