The authors 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,...,fd) of an arrangement, where fk 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 fk>(d-k+1)/kfkÅ-1 if fkℝ0. Similar results hold for spherical arrangements and oriented matroids. The authors also show that the f-vector and the h-vector of a simple arrangement is logarithmic concave, and hence unimodal.
