We generalize to the anisotropic case some classical and recent results on the (n – 1)-Minkowski content of rectifiable sets in ℝn, and on the outer Minkowski content of subsets of ℝn. In particular, a general formula for the anisotropic outer Minkowski content is provided; it applies to a wide class of sets that are stable under finite unions.

We deal with a subject in the interplay between nonparametric statistics and geometricmeasure theory. The measure L0(G) of theboundary of a set G ⊂ ℝd (withd ≥ 2) can be formally defined, via a simple limit, bythe so-called Minkowski content. We study the estimation ofL0(G) from a sample of random points insideand outside G. The sample design assumes that, for each sample point, weknow (without error) whether or not that point belongs to G. Under thisdesign we suggest a simple nonparametric estimator and investigate its consistencyproperties. The main emphasis in this paper is on generality. So we are especiallyconcerned with proving the consistency of our estimator under minimal assumptions on theset G. In particular, we establish a mild shape condition onG under which the proposed estimator is consistent inL2. Roughly speaking, such condition establishes that theset of “very spiky” points at the boundary of G must be “small”. This isformalized in terms of the Minkowski content of such set. Several examples arediscussed.

We study a nonparametric method for estimating the boundary measure of a compact body G ⊂ ℝd (the boundary length when d = 2 and the surface area for d = 3) in the case when this measure agrees with the corresponding Minkowski content. The estimator we consider is closely related to the one proposed in Cuevas, Fraiman and Rodríguez-Casal (2007). Our method relies on two sets of random points, drawn inside and outside the set G, with different sampling intensities. Strong consistency and asymptotic normality are obtained under some shape hypotheses on the set G. Some applications and practical aspects are briefly discussed.

The problem of estimating the Minkowski content L0(G) of a body G ⊂ ℝd is considered. For d = 2, the Minkowski content represents the boundary length of G. It is assumed that a ball of radius r can roll inside and outside the boundary of G. We use this shape restriction to propose a new estimator for L0(G). This estimator is based on the information provided by a random sample, taken on a square containing G, in which we know whether a sample point is in G or not. We obtain the almost sure convergence rate for the proposed estimator.