  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
$K_n$ 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 $n\to\infty$. From this
we derive the behavior of the asymptotic constant for a regular polygon in the
number of vertices.
