On k-Hulls and Related Problems

For any set X of points (in any dimension) and any $k = 1,2, \cdots $, we introduce the concept of the k-hull of X. The k-hull is the set of points p such that for any hyperplane containing p there are at least k points of X in each closed half-space determined by the hyperplane. Several computational problems related to k-hulls are studied here, including computing the k-hull and finding a point in the k-hull. Some of our algorithms are of interest in themselves because of the techniques employed; in particular, a “parametric” searching technique is used in a nontrivial way.