In between k -Sets, j -Facets, and i -Faces: (i ,j) - Partitions

Abstract

Abstract. 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 $\cap$ 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 .