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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.
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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

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