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We deal with a subject in the interplay between nonparametric statistics and geometric measure theory. The measure L 0 (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 L 0 (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 L 2 . 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.

Towards a universally consistent estimator of the Minkowski content