This paper presents the R package alphahull which implements the α-convex hull and the α-shape of a finite set of points in the plane. These geometric structures provide an informative overview of the shape and properties of the point set. Unlike the convex hull, the α-convex hull and the α-shape are able to reconstruct non-convex sets. This flexibility make them specially useful in set estimation. Since the implementation is based on the intimate relation of theses constructs with Delaunay triangulations, the R package alphahull also includes functions to compute Voronoi and Delaunay tesselations. The usefulness of the package is illustrated with two small simulation studies on boundary length estimation.

Esta es la versión de autor del artículo publicado en: This is an author produced version of a paper published in:Advances in Applied Probability 44.2 (2012): 311-329 Copyright: © 2012 Project EuclidEl acceso a la versión del editor puede requerir la suscripción del recurso Access to the published version may require subscription Motivated by set estimation problems, we consider three closely related shape conditions for compact sets: positive reach, r-convexity and rolling condition. First, the relations between these shape conditions are analyzed. Second, we obtain for the estimation of sets fulfilling a rolling condition a result of "full consistency" (i.e., consistency with respect to the Hausdorff metric for the target set and for its boundary). Third, the class of uniformly bounded compact sets whose reach is not smaller than a given constant r is shown to be a P -uniformity class (in Billingsley and Topsøe's (1967) sense) and, in particular, a Glivenko-Cantelli class. Fourth, under broad conditions, the r-convex hull of the sample is proved to be a fully consistent estimator of an rconvex support in the two-dimensional case. Moreover, its boundary length is shown to converge (a.s.) to that of the underlying support. Fifth, the above results are applied to get new consistency statements for level set estimators based on the excess mass methodology (Polonik, 1995).

The problem of estimating the Minkowski content L 0 (G) of a body G ⊂ R d 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 L 0 (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.

A test for the hypothesis of uniformity on a support S ⊂ R d is proposed. It is based on the use of multivariate spacings as those studied in Janson (1987). As a novel aspect, this test can be adapted to the case that the support S is unknown, provided that it fulfils the shape condition of λ-convexity. The consistency properties of this test are analyzed and its performance is checked through a small simulation study. The numerical problems involved in the practical calculation of the maximal spacing (which is required to obtain the test statistic) are also discussed in some detail.

The problem of estimating the surface area, L 0 , of a set G ⊂ R d has been extensively considered in several fields of research. For example, stereology focuses on the estimation of L 0 without needing to reconstruct the set G. From a more geometrical point of view, set estimation theory is interested in estimating the shape of the set. Thus, surface area estimation can be seen as a further step where the emphasis is placed on an important geometric characteristic of G. The Minkowski content is an attractive way to define L 0 that has been previously used in the literature on surface area estimation. Pateiro-López and Rodríguez-Casal [B. Pateiro-López and A. Rodríguez-Casal, Length and surface area estimation under smoothness restrictions, Adv. Appl. Prob. 40(2) (2008), pp. 348-358] proposed an estimator, L n , for L 0 under convexity type assumptions. In this paper, we obtain the L 1 -convergence rate of L n .