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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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On the cells in a stationary Poisson hyperplane mosaic

Matthias Reitzner / Rolf Schneider
Published Online: 2018-07-20 | DOI: https://doi.org/10.1515/advgeom-2018-0013

Abstract

Let X be the mosaic generated by a stationary Poisson hyperplane process in ℝd. Under some mild conditions on the spherical directional distribution of (which are satisfied if the process is isotropic), we show that with probability one the set of cells (d-polytopes) of X has the following properties. The translates of the cells are dense in the space of convex bodies. Every combinatorial type of simple d-polytopes is realized infinitely often by the cells of X. A further result concerns the distribution of the typical cell.

Keywords: Poisson process; hyperplane; polyhedra; tiling

MSC 2010: Primary 60D05; Secondary 51M20; 52C22

References

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    P. Erdös, A. Rényi, On Cantor’s series with convergent ∑1/qn. Ann. Univ. Sci. Budapest. Eötvös, Sect. Math. 2 (1959), 93–109. MR0126414 Zbl 0095.26501Google Scholar

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    A. Rényi, Wahrscheinlichkeitsrechnung. VEB Deutscher Verlag der Wissenschaften, Berlin 1977. MR0474442 Zbl 0396.60001Google Scholar

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    R. Schneider, W. Weil, Stochastic and integral geometry. Springer 2008. MR2455326 Zbl 1175.60003Google Scholar

About the article


Received: 2016-09-14

Revised: 2017-02-27

Published Online: 2018-07-20


Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0013.

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