We deal with a subject in the interplay between nonparametric statistics and geometric
measure theory. The measure L0(G) of the
boundary of a set G ⊂ ℝd (with
d ≥ 2) can be formally defined, via a simple limit, by
the so-called Minkowski content. We study the estimation of
L0(G) from a sample of random points inside
and outside G. The sample design assumes that, for each sample point, we
know (without error) whether or not that point belongs to G. Under this
design we suggest a simple nonparametric estimator and investigate its consistency
properties. The main emphasis in this paper is on generality. So we are especially
concerned with proving the consistency of our estimator under minimal assumptions on the
set G. In particular, we establish a mild shape condition on
G under which the proposed estimator is consistent in
L2. Roughly speaking, such condition establishes that the
set of “very spiky” points at the boundary of G must be “small”. This is
formalized in terms of the Minkowski content of such set. Several examples are
discussed.