In between k-sets, j-facets, and i-faces: (i, j)-partitions

Abstract

Let S be a finite set of points in general position in ℝd. We call a pair (A, B) of subsets of S an (i, j)-partition of S if | A | =i, | B | = j and there is an oriented hyperplane h with S ∩ h = A and with B the set of points from S on the positive side of h. (i, j) -Partitions generalize the notions of k-sets (these are (0, k)-partitions) and j-facets ((d, j)-partitions) of point sets as well as the notion of i-faces of the convex hull of S ((i + 1, 0)-partitions). In oriented matroid terminology, (i, j)-partitions are covectors where the number of 0's is i and the numbers of + 's is j. We obtain linear relations among the numbers of (i, j)-partitions, mainly by means of a correspondence between (i - 1)-faces of so-called k-setpolytopes on the one side and (i, j)-partitions for certain j's on the other side. We also describe the changes of the numbers of (i, j)-partitions during continuous motion of the underlying point set. This allows us to demonstrate that in dimensions exceeding 3, the vector of the numbers of k-sets does not determine the vector of the numbers of j-facets - nor vice versa. Finally, we provide formulas for the numbers of (i, j)-partitions of points on the moment curve in ℝd.