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Transactions of the American Mathematical Society

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Random points on the boundary of smooth convex bodies

Author: Matthias Reitzner
Journal: Trans. Amer. Math. Soc. 354 (2002), 2243-2278
MSC (2000): Primary 60D05, 52A22
Published electronically: February 7, 2002
MathSciNet review: 1885651
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Abstract: The convex hull of $n$ independent random points chosen on the boundary of a convex body $K \subset \mathbb{R}^d$ according to a given density function is a random polytope. The expectation of its $i$-th intrinsic volume for $i=1, \dots, d$ is investigated. In the case that the boundary of $K$ is sufficiently smooth, asymptotic expansions for these expected intrinsic volumes as $n \to \infty$ are derived.

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  • 1. Affentranger, F.: The convex hull of random points with spherically symmetric distributions. Rend. Sem. Mat. Univ. Politec. Torino 49, 359-383 (1991). MR 95f:60014
  • 2. Affentranger, F.: Aproximación aleatoria de cuerpos convexos. Publ. Mat., Barc. 36, 85-109 (1992). MR 93g:52003
  • 3. Bárány, I.: Random polytopes in smooth convex bodies. Mathematika 39, 81-92 (1992). MR 93k:52002
  • 4. Bárány, I., Buchta, C.: Random polytopes in a convex polytope, independence of shape, and concentration of vertices. Math. Ann. 297, 467-497 (1993). MR 95g:52005
  • 5. Buchta, C.: Zufällige Polyeder -- Eine Übersicht. In: Hlawka, E. (ed.) Zahlentheoretische Analysis (Lect. Notes Math., vol. 1114, pp. 1-13) Berlin Heidelberg New York Tokyo: Springer 1985. MR 90k:60024
  • 6. Buchta, C., Müller, J., Tichy, R. F.: Stochastical approximation of convex bodies. Math. Ann. 271, 225-235 (1985). MR 86g:52009
  • 7. Doetsch, G.: Handbuch der Laplace-Transformation II. Basel Stuttgart: Birkhäuser 1955. MR 18:35a
  • 8. Gruber, P. M.: Expectation of random polytopes. Manuscripta Math. 91, 393-419 (1996). MR 97m:52015
  • 9. Gruber, P. M.: Comparisons of best and random approximation of convex bodies by polytopes. Rend. Circ. Mat. Palermo, II. Ser., Suppl. 50, 189-216 (1997). MR 98m:52008
  • 10. Henrici, P.: Applied and computational complex analysis. Vol. 1. Pure and Applied Mathematics. New York: Wiley - Interscience 1974. MR 51:8378
  • 11. Hug, D.: Absolute continuity for curvature measures of convex sets. II. Math. Z. 232, 437-485 (1999). MR 2000m:52009
  • 12. Miles, R. E.: Isotropic random simplices. Adv. Appl. Prob. 3, 353-382 (1971). MR 46:8274
  • 13. Müller, J.: On the mean width of random polytopes. Probab. Theory Relat. Fields. 82, 33-37 (1989). MR 90k:60026
  • 14. Müller, J.: Approximation of a ball by random polytopes. J. Approx. Theory 63, 198-209 (1990). MR 92a:52004
  • 15. Reitzner, M.: Stochastical approximation of smooth convex bodies. Mathematika, to appear.
  • 16. Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitsth. Verw. Geb. 2, 75-84 (1963). MR 27:6190
  • 17. Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten II. Z. Wahrscheinlichkeitsth. Verw. Geb. 3, 138-147 (1964). MR 29:6392
  • 18. Sangwine-Yager, J. R.: A generalization of outer parallel sets of a convex set. Proc. Amer. Math. Soc. 123, 1559-1564 (1995). MR 92f:52008
  • 19. Santaló, L. A.: Integral geometry and geometric probability. Reading, Massachusetts: Addison-Wesley 1976. MR 55:6340
  • 20. Schneider, R.: Zur optimalen Approximation konvexer Hyperflächen durch Polyeder. Math. Ann. 256, 289-301 (1981). MR 82m:52003
  • 21. Schneider, R.: Random approximation of convex sets. J. Microscopy 151, 211-227 (1988).
  • 22. Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge: Cambridge University Press 1993. MR 94d:52007
  • 23. Schneider, R.: Discrete aspects of stochastic geometry. In: Goodman, J. E., O'Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 167-184. Boca Raton: CRC Press 1997 (CRC Press Series on Discrete Mathematics and its Applications). MR 2000j:52001
  • 24. Schütt, C.: Random polytopes and affine surface area. Math. Nachr. 170, 227-249 (1994). MR 95k:52007
  • 25. Schütt, C., Werner, E.: Random polytopes with vertices on the boundary of a convex body. C. R. Acad. Sci. Paris I, 331, 697-701 (2000). MR 2001j:60027
  • 26. Weil, W., Wieacker, J. A.: Stochastic geometry. In: Gruber, P. M., Wills, J. M. (eds.) Handbook of Convex Geometry, vol. B, pp. 1391-1438. Amsterdam London New York Tokyo: North-Holland/Elsevier 1993. MR 95a:60014
  • 27. Wieacker, J. A.: Einige Probleme der polyedrischen Approximation. Diplomarbeit, Freiburg im Breisgau 1978.
  • 28. Zähle, M.: A kinematic formula and moment measures of random sets. Math. Nachr. 149, 325-340 (1990). MR 92k:60020

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Additional Information

Matthias Reitzner
Affiliation: Institut für Analysis und Technische Mathematik, Technische Universität Wien, Wiedner Hauptstrasse 8 – 10, A-1040 Vienna, Austria
Address at time of publication: Institut für Mathematk, Universität Freiburg, Eckerstrasse 1, D-79104 Freiburg, Germany

Received by editor(s): January 19, 2001
Received by editor(s) in revised form: August 16, 2001
Published electronically: February 7, 2002
Additional Notes: Research supported, in part, by the Austrian Science Foundation (Project J1940-MAT)
Article copyright: © Copyright 2002 American Mathematical Society

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