Rich cells in an arrangement of hyperplanes
Let \(A\) denote an arrangement of hyperplanes in \(\mathbb{R}^d\). A \(d\)- dimensional cell of \(A\) is said to be rich if its boundary contains a nondegenerate (i.e., a \((d - 1)\)-dimensional) piece of each of the \(n\) hyperplanes of \(A\). The authors derive bounds on \(f(d,n)\), which is the maximum number of rich cells over all such arrangements (for \(d = 2\), even the precise number is found), and they prove that for \(n \geq d \geq 3\) the relation \[ f(d,n) = {n \choose d - 2} + O(n^{d - 3}) \] holds. Furthermore, they prove a theorem on the ``Carathéodory number'' for lines in the plane: a planar arrangement \(A\) has a rich cell provided each five of the \(n \geq 6\) lines of \(A\) form a partial arrangement containing itself a rich cell.
scientific article; zbMATH DE number 803761