We consider a stationary Poisson hyperplane process $\eta$ in $\mathbb{R}^d$ with directional distribution $\varphi$. The closures of the connected components of $\mathbb{R}^d \setminus \bigcup_{H\in\eta} H$ are polytopes. The mosaic $X_{\eta}$ is the collection of these polytopes, and we call the polytopes the cells of $X_{\eta}$. Almost surely one cell contains the origin in its interior, we call it the zero cell and denote it by $Z_0$. We investigate the distribution of the random polytope $Z_0$.

In particular we prove (under a weak assumption on $\varphi$) that there exist positive constants $c_1$ and $c_2$ such that for $n$ big enough we have

\[ (c_1 n)^{-2n/(d-1)} < \mathbb{P}( Z_0 \text{ has $n$ facets} ) < (c_2 n)^{-2n/(d-1)}. \]

It extends similar results of Calka and Hilhorst \cite{HilhorstCalka08} who gave a more precise asymptotic expansion of the above probability but only in the two dimensional and isotropic case.      

Bibliography:
H.J. Hilhorst and P. Calka. J. Stat. Phys. 132(4):627--647, 2008.