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Extremal problems for geometric probabilities involving convex bodies

Published online by Cambridge University Press:  01 July 2016

Christina Bauer  and
Rolf Schneider
Christina Bauer*
Affiliation:
Albert-Ludwigs-Universität, Freiburg i. Br.
Rolf Schneider*
Affiliation:
Albert-Ludwigs-Universität, Freiburg i. Br.
*
* Postal address for both authors: Mathematics Institut, Albert-Ludwigs-Universität, Albertstr. 23b, D-79104 Freiburg i. Br., Germany
* Postal address for both authors: Mathematics Institut, Albert-Ludwigs-Universität, Albertstr. 23b, D-79104 Freiburg i. Br., Germany
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Extract

The theory of geometric probabilities is concerned with randomly generated geometric objects. The aim is to compute probabilities of certain geometric events or distributions of random variables defined in a geometric way. Very often the computation even of simple expectations is too difficult, and one has to be satisfied with establishing estimates and, if possible, sharp inequalities. In geometric probabilities, convex sets play a prominent role, since often the convexity assumptions simplify the situation considerably. Extremal problems for geometric probabilities involving convex bodies can sometimes be attacked successfully by using suitable integral-geometric transformations and then applying classical inequalities from the geometry of convex bodies, or known methods for obtaining such inequalities. Examples of such results are the topic of this paper.

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 1 , March 1995 , pp. 20 - 34
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

The original version of this paper was presented at the International Workshop on Stochastic Geometry, Stereology and Image Analysis held at the Universidad Internacional Menendez Pelayo, Valencia, Spain on 21–24 September 1993.

References

[1] Affentranger, F. (1988) Pairs of non-intersecting random flats. Probab. Theory Rel. Fields 79, 47–50.Google Scholar
[2] Affentranger, F. (1990) Random spheres in a convex body. Arch. Math. 55, 74–81.Google Scholar
[3] Affentranger, F. (1992) Approximación aleatoria de cuerpos convexos. Publ. Mat. 36, 85–109.CrossRefGoogle Scholar
[4] Bárány, I. (1992) Random polytopes in smooth convex bodies. Mathematika 39, 81–92.Google Scholar
[5] Bárány, I. and Buchta, C. (1990) On the convex hull of uniform random points in an arbitrary d-polytope. Anz. Österr. Akad. Wiss., Math.-Nat. Kl. 25–27.Google Scholar
[6] Bárány, I. and Buchta, C. (1994) Random polytopes in a convex polytope, independence of shape, and concentration of vertices. Math. Ann. 297, 467–497.Google Scholar
[7] Bauer, C. (1992) Einschließung zufälliger Homothete konvexer Körper. Diplomarbeit, Freiburg i.Br. Google Scholar
[8] Blaschke, W. (1917, 1985) Über affine Geometrie XI: Lösung des ‘Vierpunktproblems’ von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten. Leipziger Ber. 69 (1917), 436–453. Also in: Wilhelm Blaschke, Gesammelte Werke (ed. Burau, W. et al.), Vol. 3, pp. 284–301. Thales Verlag, Essen, 1985.Google Scholar
[9] Blaschke, W. (1923) Vorlesungen über Differentialgeometrie II, Affine Differentialgeometrie . Springer-Verlag, Berlin.Google Scholar
[10] Bol, G. (1942) Zur Theorie der Eikörper. Jahresber. Deutsche Math.-Verein. 52, 250–266.Google Scholar
[11] Buchta, C. (1985) Zufällige Polyeder—Eine Übersicht. In Zahlentheoretische Analysis , ed. Hlawka, E., pp. 1–13. Lecture Notes in Mathematics 1114, Springer-Verlag, Berlin.Google Scholar
[12] Dalla, L. and Larman, D. G. (1991) Volumes of a random polytope in a convex set. In Applied Geometry and Discrete Mathematics (The Victor Klee Festschrift, eds. Gritzmann, P. and Sturmfels, B.), DIMACS Ser. Discr. Math. Theor. Comput. Sci. 4, 175–180. American Mathematical Society, Providence RI.Google Scholar
[13] Giannopoulos, A. A. (1992) On the mean value of the area of a random polygon in a plane convex body. Mathematika 39, 279–290.Google Scholar
[14] Groemer, H. (1973) On some mean values associated with a randomly selected simplex in a convex set. Pacific J. Math. 45, 525–533.Google Scholar
[15] Groemer, H. (1974) On the mean value of the volume of a random polytope in a convex set. Arch. Math. 25, 86–90.Google Scholar
[16] Groemer, H. (1982) On the average size of polytopes in a convex set. Geom. Dedicata 13, 47–62.Google Scholar
[17] Hall, G. R. (1982) Acute triangles in the n-ball. J. Appl. Prob. 19, 712–715.Google Scholar
[18] Henze, N. (1983) Random triangles in convex regions. J. Appl. Prob. 20, 111–125.Google Scholar
[19] Jung, H. (1901) Ueber die kleinste Kugel, die eine raeumliche Figur einschliesst. J. reine angew. Math. 123, 241–257.Google Scholar
[20] Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin, London.Google Scholar
[21] Knothe, H. (1937) Über Ungleichungen bei Sehnenpotenzintegralen. Deutsche Math. 2, 544–551.Google Scholar
[22] Miles, R. E. (1969) Poisson flats in Euclidean spaces, Part I: A finite number of random uniform flats. Adv. Appl. Prob. 1, 211–237.Google Scholar
[23] Miles, R. E. (1970, 1974) A synopsis of ‘Poisson flats in Euclidean spaces’. Izv. Akad. Nauk Arm. SSR, Ser. Nat. 5 (1970), 263–285. Reprinted in Stochastic Geometry , ed. Harding, E. F. and Kendall, D. G., pp. 202–227. Wiley, London, 1974.Google Scholar
[24] Miles, R. E. (1971) Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.CrossRefGoogle Scholar
[25] Pfiefer, R. E. (1989) The historical development of J. J. Sylvester's four point problem. Math. Mag. 62, 309–317.Google Scholar
[26] Pfiefer, R. E. (1990) Maximum and minimum sets for some geometric mean values. J. Theoret. Prob. 3, 169–179.Google Scholar
[27] Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.Google Scholar
[28] Santaló, L. A. (1986) On the measure of line-segments entirely contained in a convex body. In Aspects of Mathematics and its Applications , ed. Barroso, J. A., pp. 677–687. North-Holland, Amsterdam.Google Scholar
[29] Schneider, R. (1982) Random hyperplanes meeting a convex body. Z. Wahrscheinlichkeitsth. 61, 379–387.Google Scholar
[30] Schneider, R. (1985) Inequalities for random flats meeting a convex body. J. Appl. Prob. 22, 710–716.Google Scholar
[31] Schneider, R. (1988) Random approximation of convex sets. J. Microscopy 151, 211–227.Google Scholar
[32] Schneider, R. (1990) Convex geometry applied to geometric probabilities. Atti del Primo Convegno Italiano di Geometria Integrale. Rend. Sem. Mat. Messina Ser. II, 13, 37–48.Google Scholar
[33] Schneider, R. (1993) Convex Bodies: the Brunn–Minkowski Theory. Encyclopaedia of Mathematics and its Applications, 44. Cambridge University Press.Google Scholar
[34] Schneider, R. and Weil, W. (1992) Integralgeometrie. Teubner, Stuttgart.Google Scholar
[35] Schneider, R. and Wieacker, J. A. (1993) Integral geometry. In Handbook of Convex Geometry , Vol. B, ed. Gruber, P. M. and Wills, J. M., pp. 1349–1390. Elsevier, Amsterdam.Google Scholar
[36] Schöpf, P. (1977) Gewichtete Volumsmittelwerte von Simplices, welche zufällig in einem konvexen Körper des Rn gewählt werden. Monatsh. Math. 83, 331–337.Google Scholar
[37] Thomas, C. (1984) Extremum properties of the intersection densities of stationary Poisson hyperplane processes. Math. Operationsforsch. Stat., Ser. Statist. 15, 443–449.Google Scholar
[38] Weil, W. (1975) Einschachtelung konvexer Körper. Arch. Math. 16, 666–669.Google Scholar
[39] Weil, W. and Wieacker, J. A. (1993) Stochastic Geometry. In Handbook of Convex Geometry , Vol. B, eds. Gruber, P. M. and Wills, J. M., pp. 1391–1438. Elsevier, Amsterdam.CrossRefGoogle Scholar
[40] Wendel, J. G. (1962) A problem in geometric probability. Math. Scand. 11, 109–111.Google Scholar