The polytopes in a Poisson hyperplane tessellation

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