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Let $Z$ be the typical cell of a stationary Poisson hyperplane tessellationin $\mathbb{R}^d$. The distribution of the number of facets $f(Z)$ of thetypical cell is investigated. It is shown, that under a well-spread conditionon the directional distribution, the quantity$n^{\frac{2}{d-1}}\sqrt[n]{\mathbb{P}(f(Z)=n)}$ is bounded from above and frombelow. When $f(Z)$ is large, the isoperimetric ratio of $Z$ is bounded awayfrom zero with high probability.  These results rely on one hand on the Complementary Theorem which provides aprecise decomposition of the distribution of $Z$ and on the other hand onseveral geometric estimates related to the approximation of polytopes bypolytopes with fewer facets.  From the asymptotics of the distribution of $f(Z)$, tail estimates for theso-called $\Phi$ content of $Z$ are derived as well as results on theconditional distribution of $Z$ when its $\Phi$ content is large.

Cells with many facets in a Poisson hyperplane tessellation