Problem. What is maximal number of connected components on which $n$ hyperplanes divides $\mathbb{R}^m$ if they all have 1 common point.
In fact this problem was firstly stated in $\mathbb{R}^3$ and even in this case I failed. I know that there is an exact formula if planes are in general position. It has order $n^m$. For $\mathbb{R}^3$ I know construction of order 2, but is it maximum? Do you know exact formula, folks?
Hint: Every hyperplane can be associated with a couple of antipodal points on $S^{n-1}$ and every point with a Voronoi cell.