Arrangements of hemispheres and halfspaces Original Research Article

A finite set of hyperplanes passing through the origin in (d+1)-dimensional Euclidean space Ed+1 divides the unit sphere Sd into several spherical polyhedral cells. To each hyperplane, one of the two open hemispheres separated by this hyperplane is selected, and thus an arrangement of hemispheres is obtained. The weight of a face in a cell is the number of hemispheres which contain this face. The number of s-faces with weight k is denoted by fs,k. The arrangement of halfspaces may be defined in a similar way, except that the number of s-faces with weight k is denoted by gs,k. In this paper, a system of equations on fs,k (resp. gs,k) is established, which enables us to express fs,k with odd s by ft,k with even t