Skip to main content
SpringerLink
Log in

Central Limit Theorem for the Volume of Random Polytopes with Vertices on the Boundary

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

Given a convex body K with smooth boundary \(\partial K\), select a fixed number n of uniformly distributed random points from \(\partial K\). The convex hull \(K_n\) of these points is a random polytope having all its vertices on the boundary of K. The closeness of the volume of \(K_n\) to a Gaussian random variable is investigated in terms of the Kolmogorov distance by combining a version of Stein’s method with geometric estimates for the surface body of K.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Bőrőczky, K.J., Fodor, F., Hug, D.: Intrinsic volumes of random polytopes with vertices on the boundary of a convex body. Trans. Am. Math. Soc. 365(2), 785–809 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Buchta, C., Müller, J., Tichy, R.F.: Stochastical approximation of convex bodies. Math. Ann. 271(2), 225–235 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lachièze-Rey, R., Peccati, G.: New Kolmogorov Berry–Esseen bounds for functionals of binomial point processes. Ann. Appl. Probab. (accepted)

  4. Last, G., Peccati, G., Schulte, M.: Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Probab. Theory Relat. Fields 165(3–4), 667–723 (2016)

    Article  MATH  Google Scholar 

  5. Reitzner, M.: Random points on the boundary of smooth convex bodies. Trans. Am. Math. Soc. 354(6), 2243–2278 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Reitzner, M.: Random polytopes and the Efron–Stein jackknife inequality. Ann. Probab. 31(4), 2136–2166 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Reitzner, M.: Central limit theorems for random polytopes. Probab. Theory Relat. Fields 133(4), 483–507 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Reitzner, M.: Random polytopes. In: Kendall, W.S., Molchanov, I. (eds.) New Perspectives in Stochastic Geometry, pp. 45–76. Oxford University Press, Oxford (2010)

    Google Scholar 

  9. Richardson, R.M., Vu, V.H., Wu, L.: Random inscribing polytopes. Eur. J. Comb. 28(8), 2057–2071 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Richardson, R.M., Vu, V.H., Wu, L.: An inscribing model for random polytopes. Discrete Comput. Geom. 39(1–3), 469–499 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Schütt, C., Werner, E.: Polytopes with vertices chosen randomly from the boundary of a convex body. In: Milman, V.D., Schechtman, G. (eds.) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1807, pp. 241–422. Springer, Berlin (2003)

    Chapter  Google Scholar 

  12. Schütt, C., Werner, E.: Surface bodies and \(p\)-affine surface area. Adv. Math. 187(1), 98–145 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Vu, V.H.: Sharp concentration of random polytopes. Geom. Funct. Anal. 15(6), 1284–1318 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Vu, V.: Central limit theorems for random polytopes in a smooth convex set. Adv. Math. 207(1), 221–243 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I am grateful to Julian Grote (Bochum) for stimulating discussions and helpful comments. I also thank two anonymous referees for careful reading and their suggestions.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Ruhr University Bochum, 44780, Bochum, Germany

    Christoph Thäle

Authors
  1. Christoph Thäle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christoph Thäle.

Additional information

Editor in Charge: Günter M. Ziegler

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Thäle, C. Central Limit Theorem for the Volume of Random Polytopes with Vertices on the Boundary. Discrete Comput Geom 59, 990–1000 (2018). https://doi.org/10.1007/s00454-017-9862-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-017-9862-2

Keywords

Mathematics Subject Classification

Search

Navigation