Summary
On the boundary of a d-dimensional convex body a probability distribution with a positive, continuous density function g is given. The convex hull of n points chosen independently according to g is a random polytope. The asymptotic behaviour (n→∞) of the expected value of the mean width of the random polytope is determined.
References
Buchta, C.: Zufällige Polyeder-Eine Übersicht. In: Hlawka, E. (ed.) Zahlentheoretische Analysis. Lect. Notes Math., vol. 1114, pp. 1–13. Berlin Heidelberg New York: Springer 1985
Buchta, C., Müller, J.: Random polytopes in a ball. J. Appl. Probab. 21, 753–762 (1984)
Buchta, C., Müller, J., Tichy, R.F.: Stochastical approximation of convex bodies. Math. Ann. 271, 225–235 (1985)
Fejes Tóth, L.: Lagerungen in der Ebene, auf der Kugel und im Raum. Berlin Göttingen Heidelberg: Springer 1953
Gruber, P.M.: Approximation of convex bodies. In: Gruber, P.M., Wills, J.M. (eds.) Convexity and its applications, pp. 131–162. Basel: Birkhäuser 1983
McClure, D.E., Vitale, R.A.: Polygonal approximation of plane convex bodies. J. Math. Anal. Appl. 51, 326–358 (1975)
Müller, J.S.: Approximation of a ball by random polytopes. Preprint (1985)
Schneider, R.: Approximation of convex bodies by random polytopes. Aequationes Math. 32, 304–310 (1987)
Schneider, R.: Random approximation of convex sets. Proceedings of the 4th international conference on stereology and stochastic geometry, Bern 1987. J. Microscopy 151, 211–227 (1988)
Schneider, R., Wieacker, J.A.: Random polytopes in a convex body. Z. Wahrscheinlichkeitstheor. Verw. Geb. 52, 69–73 (1980)
Ziezold, H.: The mean breadth of a random polytope in a convex body. Z. Wahrscheinlichkeits-theor. Verw. Geb. 68, 121–125 (1984)
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Müller, J.S. On the mean width of random polytopes. Probab. Th. Rel. Fields 82, 33–37 (1989). https://doi.org/10.1007/BF00340011
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DOI: https://doi.org/10.1007/BF00340011
Keywords
- Probability Distribution
- Density Function
- Hull
- Stochastic Process
- Asymptotic Behaviour