On statistical properties of sets fulfilling rolling-type conditions

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, a result of "full consistency" (i.e., consistency with respect to the usual set distances, plus boundary Hausdorff-convergence) is obtained for the estimation of sets fulfilling a rolling condition. 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 Tops{\o}e'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 r-convex 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).