On the Hausdorff distance between a convex set and an interior random convex hull

Abstract

The problem of estimating an unknown compact convex set K in the plane, from a sample (X1, ⋯ , Xn) of points independently and uniformly distributed over K, is considered. Let Knm be the convex hull of the sample, Δ be the Hausdorff distance, and Δn := Δ(K, Kn). Under mild conditions, limit laws for Δn are obtained. We find sequences (an), (bn) such that (Δn - bn)/an → Λ (n → ∞), where Λ is the Gumbel (double-exponential) law from extreme-value theory. As expected, the directions of maximum curvature play a decisive role. Our results apply, for instance, to discs and to the interiors of ellipses, although for eccentricity e < 1 the first case cannot be obtained from the second by continuity. The polygonal case is also considered.