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The polytopes in a Poisson hyperplane tessellation. (English) Zbl 1482.60014

Let \(X\) be a tessellation generated by a stationary Poisson hyperplane process \(\hat X\) in \(\mathbb R^d\). Assume that the directional distribution of \(\hat X\) is supported on the whole unit sphere \( S^{d-1}\) and assigns measure zero to each great sub-sphere of \( S^{d-1}\). The main result of the paper states that with probability one, each combinatorial type of a simple \(d\)-polytope appears with positive density in X: for any simple \(d\)-polytope \(Q\) we have \begin{align*} \liminf_{n\to\infty}\frac{1}{\mathrm{vol}(nB^d)}\sum_{P\in X, P\subset nB^d}1[P\simeq Q]>0, \end{align*} where \(B^d\) is the unit ball in \(\mathbb R^d\) and \(P\simeq Q\) means that \(P\) and \(Q\) are combinatorially equivalent, that is, there is a bijection between their faces that preserves the inclusion relation.
This generalizes the result from [M. Reitzner and the author, Adv. Geom. 19, No. 2, 145–150 (2019; Zbl 1422.60023)], where the authors showed that under above assumptions, with probability one, each combinatorial type of a simple \(d\)-polytope appears infinitely often.

MSC:

60D05 Geometric probability and stochastic geometry
51M20 Polyhedra and polytopes; regular figures, division of spaces
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)

Citations:

Zbl 1422.60023
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References:

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