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

The problem of estimating an unknown compact convex set K in the plane, from a sample (X-1,...,X-n) of points independently and uniformly distributed over K, is considered. Let K-n be the convex hull of the sample, a be the Hausdorff distance, and Delta(n) := Delta(K, K-n). Under mild conditions, limit laws for Delta(n) are obtained. We find sequences (a(n)), (b(n)) such that (Delta(n) - b(n))/a(n) --> Lambda (n --> infinity), where Lambda 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.