k-sets and random hulls

We re-examine the probabilistic analysis of Clarkson and Shor [5] involvingk-sets of point sets and related structures. By studying more carefully the equations that they derive, we are able to obtain refined analysis of these quantities, which lead to a collection of interesting relationships involvingk-sets, convex hulls of random samples, and generalizations of these constructs.Work on this paper has been supported by Office of Naval Research Grant N00014-89-J-3042 and N00014-90-J-1284, by National Science Foundation Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.