Abstract
László Fejes Tóth’s fascinating book [2] demonstrates in many ways the phenomenon that figures of discrete or convex geometry that are very economical, namely solving an extremal problem of isoperimetric type, often show a high degree of symmetry. Among the examples are also planar mosaics where, for instance, an extremal property leads to the hexagonal pattern. Mosaics, or tessellations, have become increasingly important for applications. Random tessellations in two or three dimensions have been suggested as models for various real structures. We refer, e.g., to chapter 10 of the book by Stoyan, Kendall, Mecke [35] and to the book by Okabe, Boots, Sugihara, and Chiu [29] on Voronoi tessellations, which also contains a chapter on random mosaics. Apart from possible applications, random mosaics are also an interesting object of study from a purely geometric point of view. Among the results of geometric appeal that have been obtained, some concern extremal problems for (roughly speaking) expected sizes of average cells under some side condition, leading to random mosaics with high symmetry or of a very simple type, namely made up of parallelepipeds only. High symmetry here means that the distribution of the random mosaic, which is usually assumed to be translation invariant, is also invariant under rotations. Extremal problems for the sizes of average cells (not taking expectations) seem senseless at first, since extrema cannot be attained.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alexander, R., The width and diameter of a simplex, Geom. Dedicata, 6 (1977), 87–94.
Fejes Tóth, L., Reguläre Figuren, Akadémiai Kiadó, Budapest and Teubner, Leipzig (1965).
Hug, D., Reitzner, M. and Schneider, R., The limit shape of the zero cell in a stationary Poisson hyperplane tessellation, Ann. Probab., 32 (2004), 1140–1167.
Hug, D., Reitzner, M. and Schneider, R., Large Poisson-Voronoi cells and Crofton cells, Adv. Appl. Prob., 36 (2004), 667–690.
Hug, D. and Schneider, R., Large cells in Poisson Delaunay tessellations, Discrete Comput. Geom., 31 (2004), 503–514.
Hug, D. and Schneider, R., Large typical cells in Poisson-Delaunay mosaics, Rev. Roumaine Math. Pures Appl., 50 (2005), 657–670.
Hug, D. and Schneider, R., Asymptotic shapes of large cells in random tessellations, Geom. Funct. Anal., 17 (2007), 156–191.
Hug, D. and Schneider, R., Typical cells in Poisson hyperplane tessellations, Discrete Comput. Geom., 38 (2007), 305–319.
Hug, D. and Schneider, R., Large faces in Poisson hyperplane mosaics, Ann. Probab., 38 (2010), 1320–1344.
Hug, D. and Schneider, R., Faces of Poisson-Voronoi mosaics, Probab. Theory Related Fields, 151 (2011), 125–151.
Hug, D. and Schneider, R., Faces with given directions in anisotropic Poisson hyperplane mosaics, Adv. Appl. Prob., 43 (2011), 308–321.
Hug, D. and Schneider, R., Reverse inequalities for zonoids and their application, Adv. Math., 228 (2011), 2634–2646.
Kovalenko, I. N., A proof of a conjecture of David Kendall on the shape of random polygons of large area, (Russian) Kibernet. Sistem. Anal. (1997), 3–10, 187; Engl. transl., Cybernet. Systems Anal., 33 (1997), 461-467.
Kovalenko, I. N., An extension of a conjecture of D. G. Kendall concerning shapes of random polygons to Poisson Voronoï cells, in: Engel, P. et al. (eds.) Voronoï’s impact on modern science, Book I. Transl. from the Ukrainian, Kyiv: Institute of Mathematics. Proc. Inst. Math. Natl. Acad. Sci. Ukr., Math. Appl., 212 (1998), 266–274.
Kovalenko, I. N., A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons, J. Appl. Math. Stochastic Anal., 12 (1999), 301–310.
Mecke, J., Inequalities for intersection densities of superpositions of stationary Poisson hyperplane processes, in: Jensen, E. B., Gundersen, H. J. G. (eds.) Proc. Second Int. Workshop Stereology, Stochastic Geometry, pp. 115–124, Aarhus (1983).
Mecke, J., Isoperimetric properties of stationary random mosaics, Math. Nachr., 117 (1984), 75–82.
Mecke, J., On some inequalities for Poisson networks, Math. Nachr., 128 (1986), 81–86.
Mecke, J., An extremal property of random flats, J. Microsc., 151 (1988), 205–209.
Mecke, J., On the intersection density of flat processes, Math. Nachr., 151 (1991), 69–74.
Mecke, J., Inequalities for the anisotropic Poisson polytope, Adv. Appl. Prob., 27 (1995), 56–62.
Mecke, J., Inequalities for mixed stationary Poisson hyperplane tessellations, Adv. Appl. Prob., 30 (1998), 921–928.
Mecke, J., On the relationship between the 0-cell and the typical cell of a stationary random tessellation, Pattern Recognition, 32 (1999), 1645–1648.
Mecke, J. and Osburg, I., On the shape of large Crofton parallelotopes, Math. Notae, 41 (2001/02) (2003), 149–157.
Mecke, J., Schneider, R., Stoyan, D. and Weil, W., Stochastische Geometrie, DMV-Seminar 16, Birkhäuser, Basel (1990).
Mecke, J. and Thomas, C., On an extreme value problem for flat processes, Commun. Stat., Stochastic Models (2), 2 (1986), 273–280.
Miles, R. E., Random polytopes: the generalisation to n dimensions of the intervals of a Poisson process, Ph.D. Thesis, Cambridge University (1961).
Miles, R. E., A heuristic proof of a long-standing conjecture of D. G. Kendall concerning the shapes of certain large random polygons, Adv. Appl. Prob., 27 (1995), 397–417.
Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N., Spatial Tessellations; Concepts and Applications of Voronoi Diagrams, 2nd ed., Wiley, Chichester (2000).
Schneider, R., Convex Bodies: the Brunn-Minkowski Theory, Cambridge University Press, Cambridge (1993).
Schneider, R., Nonstationary Poisson hyperplanes and their induced tessellations, Adv. Appl. Prob., 35 (2003), 139–158.
Schneider, R., Weighted faces of Poisson hyperplane tessellations, Adv. Appl. Prob., 41 (2009), 682–694.
Schneider, R., Vertex numbers of weighted faces in Poisson hyperplane mosaics, Discrete Comput. Geom., 44, 599–607 (2010).
Schneider, R. and Weil, W., Stochastic and Integral Geometry, Springer, Berlin, Heidelberg (2008).
Stoyan, D., Kendall, W. S. and Mecke, J., Stochastic Geometry and Its Applications, 2nd ed., Wiley, Chichester (1995).
Tanner, R. M., Some content maximizing properties of the regular simplex, Pacific J. Math., 52 (1974), 611–616.
Wieacker, J. A., Geometric inequalities for random surfaces, Math. Nachr., 142 (1989), 73–106.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 János Bolyai Mathematical Society and Springer-Verlag
About this chapter
Cite this chapter
Schneider, R. (2013). Extremal Properties of Random Mosaics. In: Bárány, I., Böröczky, K.J., Tóth, G.F., Pach, J. (eds) Geometry — Intuitive, Discrete, and Convex. Bolyai Society Mathematical Studies, vol 24. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41498-5_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-41498-5_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41497-8
Online ISBN: 978-3-642-41498-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)