  In this work we deal with the problem of support estimation under shape
restrictions. The shape restriction we deal with is an extension of the notion
of convexity named alpha-convexity. Instead of assuming, as in the convex case,
the existence of a separating hyperplane for each exterior point we assume the
existence of a separating open ball with radius alpha. Given an alpha-convex
set S, the alpha-convex hull of independent random points in S is the natural
estimator of the set. If alpha is unknown the r_n-convex hull of the sample can
be considered. We analyze the asymptotic properties of the r_n-convex hull
estimator in the bidimensional case and obtain the convergence rate for the
expected distance in measure between the set and the estimator. The geometrical
complexity of the estimator and its dependence on r_n is also obtained via the
analysis of the expected number of vertices of the r_n-convex hull.
