The polytopes in a Poisson hyperplane tessellation

For a stationary Poisson hyperplane tessellation $X$ in ${\mathbb R}^d$, whose directional distribution satisfies some mild conditions (which hold the isotropic case, for example), it was recently shown that with probability one every combinatorial type of simple $d$-polytope is realized infinitely often by polytopes $X$. This result strengthened here: one, such appears among not only often, but positive density.