A solution to a problem of Fedorov-Grünbaum

Abstract

We determine completely the set $\mathcal{L}$ of pairs $(n,f)$ of integers for which there exist arrangements with $n$ lines and $f$ cells or, in other interpretation, for which there exists a zonohedra witn $n$ zones and $2f$ vertices. The pair $(n,f)$ is in $\mathcal{L}$ if and only if there are non-negative integers $k$ and $t$ satisfying $ t \leq \Big( \begin{matrix}k\\2 \end{matrix}\Big)$ such that $n \geq k+t+2$ and $f=(n-k)(k+1)+\Big( \begin{matrix}k\\2 \end{matrix}\Big)-t$. For each $k$ and $t$ the points $(n,f)$ satisfying these conditions are all the lattice points on a halfline $R(k,t)$, and these halflines are disjoint. Moreover, each point $(n,f) \in \mathcal{L}$ can be obtained from an easily descibed arrangement. We have found out also some characteristic properties of the arrangements corresponding to one and the same pair $(n,f) \in \mathcal{L}$.