Counting External Facets of Simple Hyperplane Arrangements

The number of external facets of a simple arrangement depends on its combinatorial type. A computation framework for counting the number of external facets is introduced and improved by exploiting the combinatorial structure of the set of sign vectors of the cells of the arrangement. 1 Background and introduction n hyperplanes in dimension d form a hyperplane arrangement. An hyperplane arrangement is called simple if n ≥ d and any d hyperplanes intersect at a unique distinct point. A facet of a hyperplane arrangement belongs to either zero, one or two bounded cells. We call a facet external if it belongs to exactly one bounded cell. It has been shown that the line arrangement (d = 2) that minimizes the number of external facets maximizes the average diameter, but it is not known whether this relation holds for general dimension. The computational results for small n and d will give us a better insight into this problem. One of the computational combinatorial problems herein is as following. Given a simple arrangement Ad,n represented by n inequalities with d variables, count its external facets. As an example, Figure 1 shows the enumeration of the 6 line arrangements formed by 5 lines, or A2,5. Among the 6 arrangements, Ao has 8 external facets, which is minimal. The star-shaped arrangement at the bottom right corner has 10 external facets and all the others have 9. Let Ad,n be a simple arrangement formed by hyperplanes h1, h2, . . . , hn. Each hyperplane partition the space into 2 sides: positive and negative. By giving each hyperplane of Ad,n an orientation indicating which side is positive, each cell is associated with a sign vector whose ith element (i = 1, . . . , n) indicates which side of hyperplane hi the cell is located on (+ and − for positive and negative side respectively). See Figure 2 for the sign vectors of all the cells of A2,4.