Discrete & Computational Geometry

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

An Inscribing Model for Random Polytopes



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 
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  1. 1.
    Alon, N., Spencer, J.: The Probabilistic Method. Wiley, New York (2000) MATHGoogle Scholar
  2. 2.
    Azuma, K.: Weighted sums of certain dependent random variables. Tohoku Math. J. 19, 357–367 (1967) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Baldi, P., Rinott, Y.: On normal approximations of distributions in terms of dependency graphs. Ann. Probab. 17(4), 1646–1650 (1989) MATHCrossRefMathSciNetGoogle 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) MATHMathSciNetCrossRefGoogle 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) MATHMathSciNetGoogle Scholar
  9. 9.
    Kim, J.H., Vu, V.H.: Concentration of multi-variate polynomials and its applications. Combinatorica 20(3), 417–434 (2000) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Reitzner, M.: Central limit theorems for random polytopes. Probab. Theory Relat. Fields 133, 483–507 (2005) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Reitzner, M.: Random polytopes and the Efron–Stein jacknife inequality. Ann. Probab. 31, 2136–2166 (2003) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Reitzner, M.: Random points on the boundary of smooth convex bodies. Trans. Am. Math. Soc. 354(6), 2243–2278 (2002) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrsch. Verw. Geb. 2, 75–84 (1963) MATHCrossRefGoogle Scholar
  14. 14.
    Rinott, Y.: On normal approximation rates for certain sums of random variables. J. Comput. Appl. Math. 55, 135–143 (1994) MATHCrossRefMathSciNetGoogle 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) MATHCrossRefMathSciNetGoogle 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

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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

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

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