BOUNDING THE NUMBER OF K-FACES IN ARRANGEMENTS OF HYPERPLANES

We study certain structural problems of arrangements of hyperplanes in d-dimensional Euclidean space. Of special interest are nontrivial relations satisfied by the f-vector f = (f0, f1, ..., f(d)) of an arrangement, where f(k) denotes the number of k-faces. The first result is that the mean number of (k - 1)-faces lying on the boundary of a fixed k-face is less than 2k in any arrangement, which implies the simple linear inequality f(k) > (d - k + 1)/kf(k - 1) if f(k) not-equal 0. Similar results hold for spherical arrangements and oriented matroids. We also show that the f-vector and the h-vector of a simple arrangement is logarithmic concave, and hence unimodal.