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

Let S be a finite set of points in general position in R-d. 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 boolean AND 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-set polytopes 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 R-d.