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Some Classical Problems in Random Geometry

  • Pierre CalkaEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2237)

Abstract

This chapter is intended as a first introduction to selected topics in random geometry. It aims at showing how classical questions from recreational mathematics can lead to the modern theory of a mathematical domain at the interface of probability and geometry. Indeed, in each of the four sections, the starting point is a historical practical problem from geometric probability. We show that the solution of the problem, if any, and the underlying discussion are the gateway to the very rich and active domain of integral and stochastic geometry, which we describe at a basic level. In particular, we explain how to connect Buffon’s needle problem to integral geometry, Bertrand’s paradox to random tessellations, Sylvester’s four-point problem to random polytopes and Jeffrey’s bicycle wheel problem to random coverings. The results and proofs selected here have been especially chosen for non-specialist readers. They do not require much prerequisite knowledge on stochastic geometry but nevertheless comprise many of the main results on these models.

Notes

Acknowledgements

The author warmly thanks two anonymous referees for their careful reading of the original manuscript, resulting in an improved and more accurate exposition.

References

  1. 1.
    D. Ahlberg, V. Tassion, A. Teixeira, Existence of an unbounded vacant set for subcritical continuum percolation (2017). https://arxiv.org/abs/1706.03053
  2. 2.
    K.S. Alexander, Finite clusters in high-density continuous percolation: compression and sphericality. Probab. Theory Relat. Fields 97, 35–63 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    R.V. Ambartzumian, A synopsis of combinatorial integral geometry. Adv. Math. 37, 1–15 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    F. Avram, D. Bertsimas, On central limit theorems in geometrical probability. Ann. Appl. Probab. 3(4), 1033–1046 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    F. Baccelli, S. Zuyev, Poisson-Voronoi spanning trees with applications to the optimization of communication networks. Oper. Res. 47, 619–631 (1999)zbMATHCrossRefGoogle Scholar
  6. 6.
    J.-E. Barbier, Note sur Ie probleme de l’aiguille et Ie jeu du joint couvert. J. Math. Pures Appl. 5, 273–286 (1860)Google Scholar
  7. 7.
    I. Bárány, Random polytopes in smooth convex bodies. Mathematika 39, 81–92 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    I. Bárány, Sylvester’s question: the probability that n points are in convex position. Ann. Probab. 27, 2020–2034 (1999)MathSciNetzbMATHGoogle Scholar
  9. 9.
    I. Bárány, A note on Sylvester’s four-point problem. Studia Sci. Math. Hungar. 38, 733–77 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Y.M. Baryshnikov, R.A. Vitale, Regular simplices and Gaussian samples. Discret. Comput. Geom. 11, 141–147 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    V. Baumstark, G. Last, Gamma distributions for stationary Poisson flat processes. Adv. Appl. Probab. 41, 911–939 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    J. Bertrand, Calcul des probabilités (Gauthier-Villars, Paris, 1889)zbMATHGoogle Scholar
  13. 13.
    T. Biehl, Über Affine Geometrie XXXVIII, Über die Schüttlung von Eikörpern. Abh. Math. Semin. Hamburg Univ. 2, 69–70 (1923)zbMATHCrossRefGoogle Scholar
  14. 14.
    H. Biermé, A. Estrade, Covering the whole space with Poisson random balls. ALEA Lat. Am. J. Probab. Math. Stat. 9, 213–229 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    W. Blaschke, Lösung des “Vierpunktproblems” von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten. Leipziger Berichte 69, 436–453 (1917)zbMATHGoogle Scholar
  16. 16.
    W. Blaschke, Vorlesungen über Differentialgeometrie II: Affine Differentialgeometrie (Springer, Berlin, 1923)zbMATHGoogle Scholar
  17. 17.
    W. Blaschke, Integralgeometrie 2: Zu Ergebnissen von M.W. Crofton. Bull. Math. Soc. Roum. Sci. 37, 3–11 (1935)Google Scholar
  18. 18.
    D. Bosq, G. Caristi, P. Deheuvels, A. Duma P. Gruber, D. Lo Bosco, V. Pipitone, Marius Stoka: Ricerca Scientifica dal 1951 al 2013, vol. III (Edizioni SGB, Messina, 2014)Google Scholar
  19. 19.
    C. Buchta, An identity relating moments of functionals of convex hulls. Discret. Comput. Geom. 33, 125–142 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    G.-L.L. Comte de Buffon, Histoire naturelle, générale et particulière, avec la description du cabinet du Roy. Tome Quatrième (Imprimerie Royale, Paris, 1777)Google Scholar
  21. 21.
    P. Bürgisser, F. Cucker, M. Lotz, Coverage processes on spheres and condition numbers for linear programming. Ann. Probab. 38, 570–604 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    P. Calka, The distributions of the smallest disks conta ining the Poisson-Voronoi typical cell and the Crofton cell in the plane. Adv. Appl. Probab. 34, 702–717 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    P. Calka, Tessellations, in New Perspectives in Stochastic Geometry, ed. by W.S. Kendall, I. Molchanov (Oxford University Press, Oxford, 2010), pp. 145–169Google Scholar
  24. 24.
    P. Calka, Asymptotic methods for random tessellations, in Stochastic Geometry, Spatial Statistics and Random Fields, ed. by E. Spodarev. Lecture Notes in Mathematics, vol. 2068 (Springer, Heidelberg, 2013), pp. 183–204Google Scholar
  25. 25.
    P. Calka, T. Schreiber, J.E. Yukich, Brownian limits, local limits and variance asymptotics for convex hulls in the ball. Ann. Probab. 41, 50–108 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    C. Carathéodory, E. Study, Zwei Beweise des Satzes daß der Kreis unter allen Figuren gleichen Umfanges den größten Inhalt hat. Math. Ann. 68, 133–140 (1910)zbMATHCrossRefGoogle Scholar
  27. 27.
    H. Carnal, Die konvexe Hülle von n rotationssymmetrisch verteilten Punkten. Z. Wahrscheinlichkeit. und verw. Gebiete 15, 168–176 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    A. Cauchy, Notes sur divers théorèmes relatifs à la rectification des courbes, et à la quadrature des surfaces. C. R. Acad. Sci. Paris 13, 1060–1063 (1841)Google Scholar
  29. 29.
    N. Chenavier, A general study of extremes of stationary tessellations with examples. Stochastic Process. Appl. 124, 2917–2953 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    P. Davy, Projected thick sections through multi-dimensional particle aggregates. J. Appl. Probab. 13, 714–722 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    S.N. Chiu, D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and its Applications, 3rd edn. Wiley Series in Probability and Statistics (Wiley, Chichester, 2013)Google Scholar
  32. 32.
    R. Cowan, The use of ergodic theorems in random geometry. Adv. Appl. Probab. 10, 47–57 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    M.W. Crofton, On the theory of local probability, applied to straight lines drawn at random in a plane; the methods used being also extended to the proof of certain new theorems in the integral calculus. Philos. Trans. R. Soc. Lond. 156, 181–199 (1868)zbMATHGoogle Scholar
  34. 34.
    D.J. Daley, Asymptotic properties of stationary point processes with generalized clusters. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 21, 65–76 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    D.J. Daley, D. Vere-Jones, An Introduction to the Theory of Point Processes. Springer Series in Statistics (Springer, New York, 1988)Google Scholar
  36. 36.
    R. Descartes, Principia Philosophiae (Louis Elzevir, Amsterdam, 1644)Google Scholar
  37. 37.
    C. Domb, Covering by random intervals and one-dimensional continuum percolation. J. Stat. Phys. 55, 441–460 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    A. Dvoretzky, On covering a circle by randomly placed arcs. Proc. Natl. Acad. Sci. U S A 42, 199–203 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    B. Efron, The convex hull of a random set of points. Biometrika 52, 331–343 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    L. Flatto, D.J. Newman Random coverings. Acta Math. 138, 241–264 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    P. Franken, D. König, U. Arndt, V. Schmidt, Queues and Point Processes (Akademie-Verlag, Berlin, 1981)zbMATHGoogle Scholar
  42. 42.
    E.N. Gilbert, Random plane networks. J. Soc. Ind. Appl. Math. 9, 533–543 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    A. Goldman, Sur une conjecture de D.G. Kendall concernant la cellule de Crofton du plan et sur sa contrepartie brownienne. Ann. Probab. 26, 1727–1750 (1998)Google Scholar
  44. 44.
    A. Goldman, The Palm measure and the Voronoi tessellation for the Ginibre process. Ann. Appl. Probab. 20, 90–128 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    S. Goudsmit, Random distribution of lines in a plane. Rev. Mod. Phys. 17, 321–322 (1945)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    J.-B. Gouéré, Subcritical regimes in the Poisson Boolean model of continuum percolation. Ann. Probab. 36, 1209–1220 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    J.-B. Gouéré, Subcritical regimes in some models of continuum percolation. Ann. Appl. Probab. 19, 1292–1318 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    H. Groemer, On some mean values associated with a randomly selected simplex in a convex set. Pac. J. Math. 45, 525–533 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    H. Hadwiger, Vorlesungen Über Inhalt, Oberfläche und Isoperimetrie (Springer, Berlin, 1957)zbMATHCrossRefGoogle Scholar
  50. 50.
    P. Hall, On continuum percolation. Ann. Probab. 13, 1250–1266 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    P. Hall, Introduction to the Theory of Coverage Processes (Wiley, New York, 1988)zbMATHGoogle Scholar
  52. 52.
    L. Heinrich, Large deviations of the empirical volume fraction for stationary Poisson grain models. Ann. Appl. Probab. 15, 392–420 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    L. Heinrich, L. Muche, Second-order properties of the point process of nodes in a stationary Voronoi tessellation. Math. Nachr. 281, 350–375 (2008). Erratum Math. Nachr. 283, 1674–1676 (2010)Google Scholar
  54. 54.
    L. Heinrich, H. Schmidt, V. Schmidt, Limit theorems for stationary tessellations with random inner cell structures. Adv. Appl. Probab. 37, 25–47 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    L. Heinrich, H. Schmidt, V. Schmidt, Central limit theorems for Poisson hyperplane tessellations. Ann. Appl. Probab. 16, 919–950 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    H.J. Hilhorst, Asymptotic statistics of the n-sided planar Poisson–Voronoi cell. I. Exact results. J. Stat. Mech. Theory Exp. 9, P09005 (2005)MathSciNetGoogle Scholar
  57. 57.
    J. Hörrmann, D. Hug, M. Reitzner, C. Thäle, Poisson polyhedra in high dimensions. Adv. Math. 281, 1–39 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    D. Hug, Random polytopes, in Stochastic Geometry, Spatial Statistics and Random Fields, ed. by E. Spodarev. Lecture Notes in Mathematics, vol. 2068 (Springer, Heidelberg, 2013), pp. 205–238Google Scholar
  59. 59.
    D. Hug, G. Last, M. Schulte, Second order properties and central limit theorems for geometric functionals of Boolean models. Ann. Appl. Probab. 26, 73–135 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    D. Hug, G. Last, W. Weil, Polynomial parallel volume, convexity and contact distributions of random sets. Probab. Theory Relat. Fields 135, 169–200 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    D. Hug, M. Reitzner, R. Schneider, The limit shape of the zero cell in a stationary Poisson hyperplane tessellation. Ann. Probab. 32, 1140–1167 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    D. Hug, R. Schneider, Asymptotic shapes of large cells in random tessellations. Geom. Funct. Anal. 17, 156–191 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    T. Huiller, Random covering of the circle: the size of the connected components. Adv. Appl. Probab. 35, 563–582 (2003)MathSciNetCrossRefGoogle Scholar
  64. 64.
    A. Hunt, R. Ewing, B. Ghanbarian, Percolation theory for flow in porous media, 3rd edn. Lecture Notes in Physics, vol. 880 (Springer, Cham, 2014)zbMATHCrossRefGoogle Scholar
  65. 65.
    S. Janson, Random coverings in several dimensions. Acta Math. 156, 83–118 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    E.T. Jaynes, The well-posed problem. Found. Phys. 3, 477–492 (1973)MathSciNetCrossRefGoogle Scholar
  67. 67.
    D.G. Kendall, Foundations of a Theory of Random Sets. Stochastic Geometry (A Tribute to the Memory of Rollo Davidson) (Wiley, London, 1974), pp. 322–376Google Scholar
  68. 68.
    M.G. Kendall, P.A.P. Moran, Geometrical Probability (Charles Griffin, London, 1963)zbMATHGoogle Scholar
  69. 69.
    J.F.C. Kingman, Random secants of a convex body. J. Appl. Probab. 6, 660–672 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    J.F.C. Kingman, Poisson Processes (Clarendon Press, Oxford, 1993)zbMATHGoogle Scholar
  71. 71.
    D.A. Klain, G.-C. Rota, Introduction to Geometric Probability (Cambridge University Press, Cambridge, 1997)zbMATHGoogle Scholar
  72. 72.
    A.N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer, Berlin, 1933)zbMATHCrossRefGoogle Scholar
  73. 73.
    G. Last, M. Penrose, Lectures of the Poisson Process (Cambridge University Press, Cambridge, 2017)zbMATHCrossRefGoogle Scholar
  74. 74.
    W. Lefebvre, T. Philippe, F. Vurpillot, Application of Delaunay tessellation for the characterization of solute-rich clusters in atom probe tomography. Ultramicroscopy 111, 200–206 (2011)CrossRefGoogle Scholar
  75. 75.
    J.-F. Marckert, The probability that n random points in a disk are in convex position. Braz. J. Probab. Stat. 31(2), 320–337 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    K.Z. Markov, C.I. Christov, On the problem of heat conduction for random dispersions of spheres allowed to overlap. Math. Models Methods Appl. Sci. 2, 249–269 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    B. Matérn, Spatial variation: Stochastic models and their application to some problems in forest surveys and other sampling investigations. Meddelanden Fran Statens Skogsforskningsinstitut, vol. 49, Stockholm (1960)Google Scholar
  78. 78.
    Matheron, G.: Random Sets and Integral Geometry. Wiley Series in Probability and Mathematical Statistics (Wiley, New York, 1975)Google Scholar
  79. 79.
    J. Mecke, Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9, 36–58 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    J. Mecke, On the relationship between the 0-cell and the typical cell of a stationary random tessellation. Pattern Recogn. 32, 1645–1648 (1999)CrossRefGoogle Scholar
  81. 81.
    R. Meesters, R. Roy, Continuum Percolation (Cambridge University Press, New York, 1996)CrossRefGoogle Scholar
  82. 82.
    J.L. Meijering, Interface area, edge length and number of vertices in crystal aggregates with random nucleation. Philips Res. Rep. 8, 270–90 (1953)zbMATHGoogle Scholar
  83. 83.
    R.E. Miles, Random polygons determined by random lines in a plane I. Proc. Natl. Acad. Sci. U S A 52, 901–907 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    R.E. Miles, Random polygons determined by random lines in a plane II. Proc. Natl. Acad. Sci. U S A 52, 1157–1160 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    R.E. Miles, The random division of space. Suppl. Adv. Appl. Probab. 4, 243–266 (1972)zbMATHCrossRefGoogle Scholar
  86. 86.
    R.E. Miles, The various aggregates of random polygons determined by random lines in a plane. Adv. Math. 10, 256–290 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  87. 87.
    R.E. Miles, Estimating aggregate and overall characteristics from thich sections by transmission microscopy. J. Microsc. 107, 227–233 (1976)CrossRefGoogle Scholar
  88. 88.
    J. Møller, Random tessellations in \({\mathbb R}^d\). Adv. Appl. Probab. 21, 37–73 (1989)Google Scholar
  89. 89.
    J. Møller, Random Johnson–Mehl tessellations. Adv. Appl. Probab. 24, 814–844 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    J. Møller, Lectures on Random Voronoi Tessellations. Lecture Notes in Statistics, vol. 87 (Springer, New York, 1994)zbMATHCrossRefGoogle Scholar
  91. 91.
    A. Müller, D. Stoyan, Comparison Methods for Stochastic Models and Risks. Wiley Series in Probability and Statistics (Wiley, Chichester, 2002)Google Scholar
  92. 92.
    W. Nagel, V. Weiss, Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv. Appl. Probab. 37, 859–883 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  93. 93.
    J. Neveu, Processus ponctuels, in École d’été de Probabilités de Saint-Flour. Lecture Notes in Mathematics, vol. 598 (Springer, Berlin, 1977), pp. 249–445zbMATHCrossRefGoogle Scholar
  94. 94.
    New Advances in Geostatistics. Papers from Session Three of the 1987 MGUS Conference held in Redwood City, California, April 13–15, 1987. Mathematical Geology, vol. 20 (Kluwer Academic/Plenum Publishers, Dordrecht, 1988), pp. 285–475Google Scholar
  95. 95.
    New Perspectives in Stochastic Geometry, ed. by W.S. Kendall, I. Molchanov (Oxford University Press, Oxford, 2010)Google Scholar
  96. 96.
    C. Palm, Intensitätsschwankungen im Fernsprechverkehr. Ericsson Technics 44, 1–189 (1943)Google Scholar
  97. 97.
    M. Penrose, On a continuum percolation model. Adv. Appl. Probab. 23, 536–556 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    M. Penrose, Non-triviality of the vacancy phase transition for the Boolean model (2017). https://arxiv.org/abs/1706.02197
  99. 99.
    R.E. Pfiefer, The historical development of J. J. Sylvester’s four point problem. Math. Mag. 62, 309–317 (1989)MathSciNetzbMATHGoogle Scholar
  100. 100.
    H. Poincaré, Calcul des probabilités (Gauthier-Villars, Paris, 1912)zbMATHGoogle Scholar
  101. 101.
    J. Quintanilla, S. Torquato, Clustering in a continuum percolation model. Adv. Appl. Probab. 29, 327–336 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  102. 102.
    C. Redenbach, On the dilated facets of a Poisson-Voronoi tessellation. Image Anal. Stereol. 30, 31–38 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  103. 103.
    M. Reitzner, Stochastical approximation of smooth convex bodies. Mathematika 51, 11–29 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  104. 104.
    M. Reitzner, Central limit theorems for random polytopes. Probab. Theory Relat. Fields 133, 483–507 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  105. 105.
    M. Reitzner, The combinatorial structure of random polytopes. Adv. Math. 191, 178–208 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  106. 106.
    M. Reitzner, Random polytopes, in New Perspectives in Stochastic Geometry, ed. by W.S. Kendall, I. Molchanov (Oxford University Press, Oxford, 2010), pp. 45–76Google Scholar
  107. 107.
    A. Rényi, R. Sulanke, Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitsth. verw. Geb. 2, 75–84 (1963)zbMATHCrossRefGoogle Scholar
  108. 108.
    A. Rényi, R. Sulanke, Über die konvexe Hülle von n zufällig gewählten Punkten. II. Z. Wahrscheinlichkeitsth. verw. Geb. 3, 138–147 (1964)zbMATHCrossRefGoogle Scholar
  109. 109.
    R. Schneider, Convex Bodies: The Brunn-Minkowski Theory (Cambridge University Press, Cambridge, 1993)zbMATHCrossRefGoogle Scholar
  110. 110.
    L.A. Santaló, Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and its Applications, vol. 1 (Addison-Wesley, Reading, 1976)Google Scholar
  111. 111.
    R. Schneider, Random hyperplanes meeting a convex body. Z. Wahrsch. Verw. Gebiete 61, 379–387 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  112. 112.
    R. Schneider, Integral geometric tools for stochastic geometry, in Stochastic Geometry, ed. by W. Weil. Lectures Given at the C.I.M.E. Summer School held in Martina Franca. Lecture Notes in Mathematics, vol. 1892 (Springer, Berlin, 2007), pp. 119–184Google Scholar
  113. 113.
    R. Schneider, W. Weil, Stochastic and Integral Geometry (Springer, Berlin, 2008)zbMATHCrossRefGoogle Scholar
  114. 114.
    R. Schneider, J.A. Wieacker, Random polytopes in a convex body. Z. Wahrsch. Verw. Gebiete 52, 69–73 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    J. Serra, Image Analysis and Mathematical Morphology (Academic, London, 1984)Google Scholar
  116. 116.
    L.A. Shepp, Covering the circle with random arcs. Isr. J. Math. 11, 328–345 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  117. 117.
    A.F. Siegel, Random space filling and moments of coverage in geometrical probability. J. Appl. Probab. 15, 340–355 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  118. 118.
    A.F. Siegel, L. Holst, Covering the circle with random arcs of random sizes. J. Appl. Probab. 19, 373–381 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  119. 119.
    H. Solomon, Geometric Probability. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 28 (SIAM, Philadelphia, 1978)Google Scholar
  120. 120.
    J. Steiner, Über parallele Flächen. Monatsber. Preuss. Akad. Wiss., Berlin (1840), pp. 114–118Google Scholar
  121. 121.
    W.L. Stevens, Solution to a geometrical problem in probability. Ann. Eugenics 9, 315–320 (1939)MathSciNetCrossRefGoogle Scholar
  122. 122.
    D. Stoyan, Applied stochastic geometry: a survey. Biometrical J. 21, 693–715 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  123. 123.
    J.J. Sylvester, Problem 1491. The Educational Times, London (April, 1864)Google Scholar
  124. 124.
    P. Valtr, Probability that n random points are in convex position. Discret. Comput. Geom. 13, 637–643 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  125. 125.
    P. Valtr, The probability that n random points in a triangle are in convex position. Combinatorica 16, 567–573 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  126. 126.
    V.H. Vu, Sharp concentration of random polytopes. Geom. Funct. Anal. 15, 1284–1318 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  127. 127.
    W. Weil, Point processes of cylinders, particles and flats. Acta Appl. Math. 9, 103–136 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  128. 128.
    V. Weiss, R. Cowan, Topological relationships in spatial tessellations. Adv. Appl. Probab. 43, 963–984 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  129. 129.
    J.G. Wendel, A problem in geometric probability. Math. Scand. 11, 109–111 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  130. 130.
    W.A. Whitworth, Choice and Chance (D. Bell, Cambridge, 1870)zbMATHGoogle Scholar
  131. 131.
    F. Willot, D. Jeulin, Elastic behavior of composites containing Boolean random sets of inhomogeneities. Int. J. Eng. Sci. 47, 313–324 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.University of RouenLMRSSaint-Étienne-du-RouvrayFrance

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