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ON THE MONOTONICITY OF THE EXPECTED VOLUME OF A RANDOM SIMPLEX

Part of: Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  20 June 2011

Luis Rademacher
Luis Rademacher*
Affiliation:
School of Computer Science and Engineering, The Ohio State University, Columbus, OH 43210, U.S.A. (email: lrademac@cse.ohio-state.edu)
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Abstract

Consider a random simplex in a d-dimensional convex body which is the convex hull of d+1 random points from the body. We study the following question: as a function of the convex body, is the expected volume of such a random simplex monotone non-decreasing under inclusion? We show that this is true when d is 1 or 2, but does not hold for d≥4. We also prove similar results for higher moments of the volume of a random simplex, in particular for the second moment, which corresponds to the determinant of the covariance matrix of the convex body. These questions are motivated by the slicing conjecture.

MSC classification

Secondary: 60D05: Geometric probability and stochastic geometry 52A23: Asymptotic theory of convex bodies
Type
Research Article
Information
Mathematika , Volume 58 , Issue 1 , January 2012 , pp. 77 - 91
Copyright
Copyright © University College London 2012

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