Cells with many facets in arrangements of hyperplanes (Q1185096)

Let \({\mathcal H}\) be an euclidean \(d\)-arrangement of hyperplanes, i.e., a finite collection of hyperplanes in \(E^ d\) where no point belongs to all the hyperplanes of \({\mathcal H}\). (Similarly one can consider real projective space \(\mathbb{P}^ d\).) Such an arrangement decomposes \(E^ d\) into a \(d\)-dimensional cell complex \({\mathcal H}\). The cells of \({\mathcal H}\) are the \(d\)-cells of \({\mathcal H}\) and the facets of \({\mathcal H}\) are the \((d- 1)\)-cells of \({\mathcal H}\). If \(n\) is the number of hyperplanes of \({\mathcal H}\), then \(c\) is a complete cell of \({\mathcal H}\) if \(c\) has exactly \(n\) facets. \textit{H. Edelsbrunner} and \textit{D. Haussler} [Discrete Math. 60, 139-146 (1986; Zbl 0596.51011)] showed that for every \(n\geq 4\) there is an euclidean 3-arrangement of \(n\) planes having 5 complete cells and conjectured that an arrangement of sufficiently many planes in \(E^ 3\) cannot have 6 complete cells. The author here disproves the conjecture by showing that for all \(d\), \(n\) such that \(2\leq d\leq n-1\), there exists an arrangement of \(n\) hyperplanes with at least \(\sum^{d- 2}_{i=0}{n\choose i}\) complete cells. Further, the author shows that the average number of facets in the cells of an arrangement of \(n\) pseudohyperplanes in \(\mathbb{P}^ d\) is always less than \(2d+1\). The bound of \(2d+1\) can be reduced to \(2d\), \(d\geq 2\), in the case of simple arrangements, i.e. where no \(d+1\) hyperplanes have a point in common (general position).