Asymptotic efficiency in estimation of a convex set

The estimation problem is considered for closed convex set G on a plane using a sample from uniform distribution on the set. G is assumed to belong to the class of all closed convex subsets of a given circle with Lebeg measure being separated from zero or the less class of all convex sets having smooth boundaries with curvature radius ≥R₀. Asymptotics of minimax risk on these classes is studied for the case when distance between estimator and true set is measured in Hausdorff metric. Asymptotics of minimax risks is proved to be different for the considered classes. Moreover, for the class of smooth convex sets G the accurate estimators are obtained for minimax risk with the ratio of the lower boundary to the upper one ≈0.96.