Let A1,...,An be distinct k-dimensional vectors. The authors consider the problem of partitioning these vectors into m sets so as to maximize an objective which is a quasi-convex function of the sum of vectors in each set. They show that there exists an optimal partition whose sets have (pairwise) disjoint conic hulls. The authors also show that if the number of vectors in each of the sets is constrained, then a weaker conclusion hold, namely, there exists an optimal partition whose sets have (pairwise) disjoint convex hulls. The results rely on deriving necessary and sufficient conditions for a point to be an extreme point of a corresponding polytope.
