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Random convex hulls in a product of balls
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  • Published: December 1990

Random convex hulls in a product of balls

  • Rex A. Dwyer1 

Probability Theory and Related Fields volume 86, pages 457–467 (1990)Cite this article

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Summary

The convex hull of a set of points sampled independently and uniformly from the Cartesian product of balls of various dimensions is investigated. Bounds on the asymptotic behavior of the expected combinatorial complexity volume, and mean width are derived when the distribution is held fixed and the sample size approaches infinity. The expected combinational complexity and volume are found to depend (up to constant factors) only on the greatest dimension of any factor ball and the number of balls of that dimension. On the other hand, the expected mean width depends only on the number of balls and the dimensions of the product.

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References

  1. Affentranger, F., Wieacker, J.A.: On the convex hull of uniform random points in a simpled-polytope. Discrete Comput. Geom. (to appear)

  2. Bárány, I.: Intrinsic volumes andf-vectors of random polytopes Math. Ann.285, 671–699 (1989)

    Google Scholar 

  3. Bárány, I., Larman, D.C. Convex bodies, economic cap coverings, random polytopes. Mathematika35, 274–291 (1988)

    Google Scholar 

  4. Bentley, J.L., Kung, H.T., Schkolnick, M., Thompson, C.D.: On the average number of maxima in a set of vectors. J. Assoc. Comput. Mach.25, 536–543 (1978)

    Google Scholar 

  5. Brøndsted, A.: An introduction to convex polytopes. Berlin Heidelberg New York: Springer 1983

    Google Scholar 

  6. Buchta, C.: Zufällige Polyeder: eine Übersicht. In: Halwaka, E. (ed.) Zahlentheoretische Analysis. (Lect. Notes Math., vol. 1114, pp. 1–13). Berlin Heidelberg New York: Springer 1985

    Google Scholar 

  7. Devroye, L.P.: A note on finding convex hulls via maximal vectors. Inf. Process. Lett.11, 53–56 (1980)

    Google Scholar 

  8. Devroye, L.P.: How to reduce the average complexity of convex hull finding algorithms. Comput. Math. Appl.7, 299–308 (1981)

    Google Scholar 

  9. Dwyer, R.A.: On the convex hull of random points in a polytope. J. Appl. Probab.25, 688–699 (1988)

    Google Scholar 

  10. Dwyer, R.A.: Convex hulls of samples from spherically symmetric distributions. Discrete Appl. Math. (to appear)

  11. Efron, B.: The convex hull of a random set of points. Biometrika52, 331–342 (1965)

    Google Scholar 

  12. Raynaud, H.: Sur l'envelope convexe des nuages des points aléatoires dans ℝn, I. J. Appl. Probab.7, 35–48 (1970)

    Google Scholar 

  13. Schneider, R.: Approxiamtion of convex bodies by random polytopes. AEquationes Math.32, 304–310 (1987)

    Google Scholar 

  14. Schneider, R.: Approximation of convex sets. J. Microscopy151, 211–227 (1988)

    Google Scholar 

  15. Schneider, R., Wieacker, J.A.: Random polytopes in a convex body. Z. Wahrscheinlichkeitstheor. Verw. Geb.52, 69–73 (1980)

    Google Scholar 

  16. Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge: Cambridge University Press 1927

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, North Carolina State University, 27695-8206, Ralgeigh, NC, USA

    Rex A. Dwyer

Authors
  1. Rex A. Dwyer
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Additional information

Supported by the National Science Foundation under Grants CCR-8658139 and CCR-8908782

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Cite this article

Dwyer, R.A. Random convex hulls in a product of balls. Probab. Th. Rel. Fields 86, 457–467 (1990). https://doi.org/10.1007/BF01198169

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  • Received: 08 September 1989

  • Revised: 09 March 1990

  • Issue Date: December 1990

  • DOI: https://doi.org/10.1007/BF01198169

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Keywords

  • Hull
  • Stochastic Process
  • Asymptotic Behavior
  • Probability Theory
  • Convex Hull
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