Journal of Theoretical Probability

, Volume 3, Issue 3, pp 387–393 | Cite as

Distribution-independent properties of the convex hull of random points

  • Christian Buchta
Article

Abstract

Denote byVn(d) the expected volume of the convex hull ofn points chosen independently according to a given probability measure μ in Euclideand-spaceEd. Ifd=2 ord=3 and μ is the measure corresponding to the uniform distribution on a convex body inEd, Affentranger and Badertscher derived that
$$V_{d + 2m}^{(d)} = \sum\limits_{k = 1}^m {(2^{2k} - 1)} \frac{{B_{2k} }}{k}\left( {\begin{array}{*{20}c} {d + 2m} \\ {2k - 1} \\ \end{array} } \right)V_{d + 2m - 2k + 1}^{(d)} \left( {m = 1,2, \ldots } \right)$$
where the constantsB2k are the Bernoulli numbers. We verify this identity for arbitrary measures μ in spaces of arbitrary dimensiond. The extended version of the identity follows from a more general theorem, which is proved by constructing a moment functionalL such that
$$V_{d + 1 + p} = \left( {\begin{array}{*{20}c} {d + 1 + p} \\ {d + 1} \\ \end{array} } \right)L\left[ {x^p + \left( {1 - x} \right)^p } \right]\left( {p = 0,1, \ldots } \right)$$

Key Words

Random points random polytope convex hull geometric probability distribution-independent properties moment functional Bernoulli numbers 

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References

  1. 1.
    Affentranger, F. (1988). The expected volume of a random polytope in a ball.J. Microsc. 151, 277–287.Google Scholar
  2. 2.
    Affentranger, F. (1988). Generalization of a formula of C. Buchta about the convex hull of random points.Elem. Math. 43, 39–45, 151–152.Google Scholar
  3. 3.
    Affentranger, F., and Wieacker, J. A. (1990). On the convex hull of uniform random points in a simpled-polytope.Discrete Comput. Geom. (in press).Google Scholar
  4. 4.
    Badertscher, E. (1989). An explicit formula about the convex hull of random points.Elem. Math. 44, 104–106.Google Scholar
  5. 5.
    Bárány, I. (1989). Intrinsic volumes andf-vectors of random polytopes.Math. Ann. 285, 671–699.Google Scholar
  6. 6.
    Berger, G., and Tasche, M. (1988). Hermite-Lagrange interpolation and Schur's expansion of sin πx.J. Approx. Theory 53, 17–25.Google Scholar
  7. 7.
    Buchta, C. (1984). Das Volumen von Zufallspolyedern im Ellipsoid.Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 1984, 1–4.Google Scholar
  8. 8.
    Buchta, C. (1984). Zufallspolygone in konvexen Vielecken.J. Reine Angew. Math. 347, 212–220.Google Scholar
  9. 9.
    Buchta, C. (1985). Zufällige Polyeder — Eine Übersicht. In Hlawka, E. (ed.),Zahlentheoretische Analysis, Lecture Notes in Mathematics, Vol. 1114, Springer-Verlag, Berlin, pp. 1–13.Google Scholar
  10. 10.
    Buchta, C. (1986). On a conjecture of R. E. Miles about the convex hull of random points.Monatsh. Math. 102, 91–102.Google Scholar
  11. 11.
    Fisher, L. (1971). Limiting convex hulls of samples: theory and function space examples.Z. Wahrsch. Verw. Gebiete 18, 281–297.Google Scholar
  12. 12.
    Knopp, K. (1922).Theorie und Anwendung der unendlichen Reihen, Springer-Verlag, Berlin.Google Scholar
  13. 13.
    Rényi, A., and Sulanke, R. (1964). Über die konvexe Hülle vonn zufällig gewählten Punkten. II.Z. Wahrsch. Verw. Gebiete 3, 138–147.Google Scholar
  14. 14.
    Schneider, R. (1988). Random approximation of convex sets.J. Microsc. 151, 211–227.Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • Christian Buchta
    • 1
  1. 1.Institut für Analysis, Technische Mathematik und VersicherungsmathematikTechnische UniversitätViennaAustria

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