  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 Topsoe'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).
