Random approximation of convex bodies in Hausdorff metric

While there is extensive literature on approximation, deterministic as well as random, of general convex bodies K in the symmetric difference metric, or other metrics arising from intrinsic volumes, very little is known for corresponding random results in the Hausdorff distance when the approximant Kₙ is given by the convex hull of n independent random points chosen uniformly on the boundary or in the interior of K. When K is a polygon and the points are chosen on its boundary, we determine the exact limiting behavior of the expected Hausdorff distance between a polygon as n. From this we derive the behavior of the asymptotic constant for a regular polygon in the number of vertices.