Arrangements of oriented hyperplanes (Q1312195)

The paper refers to arrangements of \(n\) oriented hyperplanes in \(E^ d\). For \(n\) and \(d\) given, the author derives an upper bound on the number \(c_ k\) of convex cells which are covered by precisley \(k\) half-spaces. Denoting the corresponding maximal number by \(C_ k(n,d)\), for \(n>d\) the following recursive inequality holds: \[ C_ k(n,d) \leq [n/(d+1)] (C_ k(n-1,d-1)+C_{k-1} (n-1,d-1))+[1/(d+1)] S_ k(n,d-1), \] with \(C_ 0(n,d)=1\), \(C_ k (n,1)=k+1\) for \(0 \leq k \leq n/2\), and \[ S_ k(n,d- 1)=[n/(d+1)] (S_ k(n-1,d-1)+S_{k-1}(n-1,d-1)). \] The latter number refers to pairs of respective cells in spherical arrangements, i.e. of cells on \(S^{d-1}\) generated by intersections of \(S^{d-1}\) and arrangements of \((d-1)\)-dimensional subspaces of \(E^ d\).