Discrete & Computational Geometry

, Volume 6, Issue 3, pp 291–305 | Cite as

On the convex hull of uniform random points in a simpled-polytope

  • Fernando Affentranger
  • John A. Wieacker
Article

Abstract

Denote the expected number of facets and vertices and the expected volume of the convex hullPn ofn random points, selected independently and uniformly from the interior of a simpled-polytope byEn(f), En(v), andEn(V), respectively. In this note we determine the sharp constants of the asymptotic expansion ofEn(f), En(v), andEn(V), asn tends to infinity. Further, some results concerning the expected shape ofPn are given.

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References

  1. 1.
    I. Bárány, Intrinsic volumes andf-vectors of random polytopes,Math. Ann. 285 (1989), 671–699.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    I. Bárány and D. G. Larman, Convex bodies, economic cap coverings, random polytopes,Mathematika 35 (1988), 274–291.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    A. Brøndsted,An introduction to Convex Polytopes, Springer-Verlag, New York, 1982.Google Scholar
  4. 4.
    C. Buchta, Stochastische Approximation konvexer Polygone,Z. Warsch. Verw. Gebiete 67 (1984), 283–304.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    C. Buchta, Zufällige Polyeder—Eine Übersicht, inZahlentheoretische Analysis (ed. by E. Hlawka), Lecture Notes in Mathematics, Vol. 1114, Springer-Verlag, Berlin, 1985, pp. 1–13.CrossRefGoogle Scholar
  6. 6.
    C. Buchta, A note on the volume of a random polytope in a tetrahedron,Illinois J. Math. 30 (1986), 653–659.MathSciNetMATHGoogle Scholar
  7. 7.
    C. Buchta, A remark on random approximation of simple polytopes,Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. (1989), 17–20.Google Scholar
  8. 8.
    R. A. Dwyer, On the convex hull of random points in a polytope,J. Appl. Probab. 25 (1988), 688–699.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    B. Efron, The convex hull of a random set of points,Biometrika 52 (1965), 331–343.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    W. Feller,An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn., Wiley, New York, 1971.MATHGoogle Scholar
  11. 11.
    H. Groemer, On some mean values associated with a randomly selected simplex in a convex set,Pacific J. Math. 45 (1973), 525–533.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    V. Klee, What is the expected volume of a simplex whose vertices are chosen at random from a given convex body?Amer. Math. Monthly 76 (1969), 286–288.MathSciNetCrossRefGoogle Scholar
  13. 13.
    W. J. Reed, Random points in a simplex,Pacific J. Math. 54 (1974), 183–198.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    A. Rényi and R. Sulanke, Über die konvexe Hülle vonn zufällig gewählten Punkten,Z. Wahrsch. Verw. Gebiete 2 (1963), 75–84.CrossRefMATHGoogle Scholar
  15. 15.
    A. Rényi and R. Sulanke, Über die konvexe Hülle vonn zufällig gewählten Punkten, II,Z. Wahrsch. Verw. Gebiete 3 (1964), 138–147.CrossRefMATHGoogle Scholar
  16. 16.
    L. A. Santaló,Integral Geometry and Geometric Probability, Addison-Wesley, Reading, Massachusetts, 1976.MATHGoogle Scholar
  17. 17.
    R. Schneider, Random approximation of convex sets,J. Microscopy 151 (1988), 211–227.CrossRefGoogle Scholar
  18. 18.
    B. F. van Wel, The convex hull of a uniform sample from the interior of a simpled-polytope,J. Appl. Probab. 27 (1989), 259–273.CrossRefGoogle Scholar
  19. 19.
    E. T. Whittaker and G. N. Watson,A Course of Modern Analysis, 4th edn., Cambridge University Press, Cambridge, 1952.Google Scholar
  20. 20.
    J. A. Wieacker, Einige Probleme der polyedrischen Approximation, Diplomarbeit, Freiburg i. Br., 1978.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1991

Authors and Affiliations

  • Fernando Affentranger
    • 1
  • John A. Wieacker
    • 1
  1. 1.Mathematisches InstitutAlbert-Ludwigs-UniversitätFreiburg i. Br.Germany

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