Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T22:05:57.242Z Has data issue: false hasContentIssue false
  • English
  • Français

On the mean value of the area of a random polygon in a plane convex body

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Apostolos A. Giannopoulos
Apostolos A. Giannopoulos
Affiliation:
Department of Mathematics, University of Crete, Iraklion, Crete, Greece.
Get access

Extract

Let K be a convex body in Euclidean space Rd, d≥2, with volume V(K) = 1, and n ≥ d +1 be a natural number. We select n independent random points y1, y2, …, yn from K (we assume they all have the uniform distribution in K). Their convex hull co {y1, y2, …, yn} is a random polytope in K with at most n vertices. Consider the expected value of the volume of this polytope

It is easy to see that if U: Rd → Rd is a volume preserving affine transformation, then for every convex body K with V(K) = 1, m(K, n) = m(U(K), n).

MSC classification

Secondary: 52A22: Random convex sets and integral geometry
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 279 - 290
Copyright
Copyright © University College London 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Blaschke, W.. (1917). Losung des “Vierpunktproblems” von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten. Ber. Verh. sachs. Acad. Wiss., Math. Phys. KL., (69), 436–453.Google Scholar
2.Blaschke, W.. (1923). Vorlesungen uber Differentialgeometrie II (Springer, Berlin).Google Scholar
3.Groemer, H.. (1973). On some mean values associated with a randomly selected simplex in a convex set. Pacific J. Math., (45), 525–533.CrossRefGoogle Scholar
4.Groemer, H.. (1974). On the mean value of the volume of a random polytope in a convex set. Arch. Math., (25), 86–90.CrossRefGoogle Scholar
5.Dalla, L. and Larman, D. G.. (1991). Volumes of a random polytope in a convex set. Applied geometry and discrete mathematics. Discrete Math. Theoret Comput. Sci., (4), (Amer. Math. Soc), 175–180.Google Scholar
6.Schneider, R.. (1988). Random approximation of convex sets. Proceedings of the 4th international conference on stereology and stochastic geometry, Bern 1987. J Microscopy, (151), 211–227.CrossRefGoogle Scholar
7.John, F.. (1948). Extremum problems with inequalities as subsidiary conditions. Courant Anniv. Volume, (1948), 187–204.Google Scholar
8.Buchta, C.. (1984). Zufallspolygons in konvexen Vielecken. J. für reine u. angew. Math., (347), 22–220.Google Scholar