Counting facets and incidences
The authors show a tight bound on the maximum number of facets bounding \(n^{d-2}\leq m\leq n^ d\) distinct cells in arrangements on \(n\) hyperplanes in \(R^ d\). This yields a tight bound on the number of incidences between \(n\) hyperplanes in \(R^ d\) and \(m\) vertices in their arrangement. The same technique is employed to obtain a tight bound on the number of faces bounding all nonconvex cells in arrangements of triangles in \(R^ 3\). Some open questions are given.
scientific article; zbMATH DE number 54914