Probability Theory and Related Fields

, Volume 77, Issue 2, pp 231–240 | Cite as

On the shape of the convex hull of random points

  • Imre Bárány
  • Zoltán Füredi
Article

Summary

Denote by En the convex hull of n points chosen uniformly and independently from the d-dimensional ball. Let Prob(d, n) denote the probability that En has exactly n vertices. It is proved here that Prob(d, 2d/2d)→1 and Prob(d, 2d/2d(3/4)+ɛ)→0 for every fixed ɛ>0 when d→∞. The question whether En is a k-neighbourly polytope is also investigated.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bárány, I., Füredi, Z.: Approximation of the sphere by polytopes having few vertices. To appear in Proc. Am. Math. Soc.Google Scholar
  2. 2.
    Blaschke, W.: Vorlesungen über Differentialgeometrie II. Affine Differentialgeometrie. Berlin: Springer 1923Google Scholar
  3. 3.
    Buchta, C.: On a conjecture of R.E. Miles about the convex hull of random points. Monatsh. Math. 102, 91–102 (1986)Google Scholar
  4. 4.
    Buchta, C., Müller, J.: Random polytopes in a ball. J. Appl. Probab. 21, 753–762 (1984)Google Scholar
  5. 5.
    Chernoff, H.: A measure of asymptotic efficiency for tests of an hypothesis based on the sum of observations. Ann. Math. Statist. 23, 493–507 (1952)Google Scholar
  6. 6.
    Elekes, G.: A geometric inequality and the complexity of computing volume. Discrete. Comput. Geom. 1, 289–292 (1986)Google Scholar
  7. 7.
    Erdös, P., Spencer, J.: Probabilistic methods in combinatorics. Budapest: Akadémiai Kiadó 1974Google Scholar
  8. 8.
    Füredi, Z.: Random polytopes in the d-dimensional cube. Discrete Comput. Geom. 1, 315–319 (1986)Google Scholar
  9. 9.
    Gruber, P.M.: Approximation of convex bodies by polytopes. In: Gruber, P.M., Wills, J.M. (eds.). Convexity and its applications, pp. 131–162. Basel: Birkhauser 1983Google Scholar
  10. 10.
    Hostinsky, B.: Sur les probabilités géométriques. Publ. Fac. Sci. Univ. Masaryk. Brno. 1925Google Scholar
  11. 11.
    Kingman, J.F.C.: Random secants of a convex body. J. Appl. Probab. 6, 660–672 (1969)Google Scholar
  12. 12.
    Lovász, L.: An algorithmic theory of numbers, graphs and convexity. Preprint, Report No. 85368-OR, University of Bonn 1985Google Scholar
  13. 13.
    Miles, R.E.: Isotropic random simplices. Adv. Appl. Probab. 3, 353–382 (1971)Google Scholar
  14. 14.
    Rényi, A.: Probability theory. Amsterdam: North-Holland 1970Google Scholar
  15. 15.
    Rényi, A., Sulanke, R.: Über die convexe Hülle von n zufällig gewählten Punkten I–II. Z. Wahrscheinlichkeitstheor. Verw. Geb. 2, 75–84 (1963); 3, 138–147 (1964)Google Scholar
  16. 16.
    Spencer, F.: Sequences with small discrepancy relative to n events. Compositio Math. 47, 365–392 (1982)Google Scholar
  17. 17.
    Wendel, J.G.: A problem in geometric probability. Math. Scand. 11, 109–111 (1962)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Imre Bárány
    • 1
  • Zoltán Füredi
    • 2
  1. 1.School of OR & IECornell UniversityIthacaUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations