Discrete & Computational Geometry

, Volume 39, Issue 1–3, pp 469–499 | Cite as

An Inscribing Model for Random Polytopes

  • Ross M. RichardsonEmail author
  • Van H. Vu
  • Lei Wu


For convex bodies K with \(\mathcal {C}^{2}\) boundary in ℝ d , we explore random polytopes with vertices chosen along the boundary of K. In particular, we determine asymptotic properties of the volume of these random polytopes. We provide results concerning the variance and higher moments of this functional, as well as an analogous central limit theorem.


Convex Hull Central Limit Theorem Convex Body Random Point Discrete Comput Geom 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Alon, N., Spencer, J.: The Probabilistic Method. Wiley, New York (2000) zbMATHGoogle Scholar
  2. 2.
    Azuma, K.: Weighted sums of certain dependent random variables. Tohoku Math. J. 19, 357–367 (1967) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Baldi, P., Rinott, Y.: On normal approximations of distributions in terms of dependency graphs. Ann. Probab. 17(4), 1646–1650 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bárány, I.: Personal conversations, UCSD (2005) Google Scholar
  5. 5.
    Bárány, I.: Convex bodies, random polytopes, and approximation. In: Weil, W. (ed.) Stochastic Geometry. Springer (2005) Google Scholar
  6. 6.
    Bárány, I., Larman, D.: Convex bodies, economic cap coverings, random polytopes. Mathematika 35(2), 274–291 (1988) zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bárány, I., Reitzner, M.: Central limit theorem for random polytopes in convex polytopes. Manuscript (2005) Google Scholar
  8. 8.
    Efron, B.: The convex hull of a random set of points. Biometrika 52, 331–343 (1965) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Kim, J.H., Vu, V.H.: Concentration of multi-variate polynomials and its applications. Combinatorica 20(3), 417–434 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Reitzner, M.: Central limit theorems for random polytopes. Probab. Theory Relat. Fields 133, 483–507 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Reitzner, M.: Random polytopes and the Efron–Stein jacknife inequality. Ann. Probab. 31, 2136–2166 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Reitzner, M.: Random points on the boundary of smooth convex bodies. Trans. Am. Math. Soc. 354(6), 2243–2278 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrsch. Verw. Geb. 2, 75–84 (1963) zbMATHCrossRefGoogle Scholar
  14. 14.
    Rinott, Y.: On normal approximation rates for certain sums of random variables. J. Comput. Appl. Math. 55, 135–143 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Schneider, R.: Discrete aspects of stochastic geometry. In: Goodman, J., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 255–278. CRC Press, Boca Raton (2004) Google Scholar
  16. 16.
    Schütt, C., Werner, E.: Polytopes with vertices chosen randomly from the boundary of a convex body. In: Geometric Aspects of Functional Analysis 2001–2002. Lecture Notes in Mathematics, vol. 1807, pp. 241–422. Springer, New York (2003) Google Scholar
  17. 17.
    Sylvester, J.J.: Question 1491. Educational Times. London (April, 1864) Google Scholar
  18. 18.
    Vu, V.H.: Sharp concentration of random polytopes. Geom. Funct. Anal. 15, 1284–1318 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Vu, V.H.: Central limit theorems for random polytopes in a smooth convex set. Adv. Math. 207, 221–243 (2005) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Weil, W., Wieacker, J.: Stochastic geometry. In: Gruber, P., Wills, J. (eds.) Handbook of Convex Ceometry, vol. B, pp. 1391–1438. North-Holland, Amsterdam (1993) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of MathematicsUCSDLa JollaUSA
  2. 2.Department of MathematicsRutgers UniversityPiscatawayUSA

Personalised recommendations