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Discrete & Computational Geometry

, Volume 59, Issue 1, pp 165–174 | Cite as

Random Approximation of Convex Bodies: Monotonicity of the Volumes of Random Tetrahedra

  • Stefan Kunis
  • Benjamin Reichenwallner
  • Matthias Reitzner
Article

Abstract

Choose uniform random points \(X_1, \dots , X_n\) in a given convex set and let \({\text { conv}}[X_1, \dots , X_n]\) be their convex hull. It is shown that in dimension three the expected volume of this convex hull is in general not monotone with respect to set inclusion. This answers a question by Meckes in the negative. The given counterexample is formed by uniformly distributed points in the three-dimensional tetrahedron together with a small perturbation of it. As side result we obtain an explicit formula for all even moments of the volume of a random simplex which is the convex hull of three uniform random points in the tetrahedron and the center of one facet.

Keywords

Approximation of convex sets Random convex hull Extreme points Random simplex Sample range 

Mathematics Subject Classification

Primary 52A22 Secondary 60D05 

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Stefan Kunis
    • 1
  • Benjamin Reichenwallner
    • 2
  • Matthias Reitzner
    • 1
  1. 1.Institut für MathematikUniversität OsnabrückOsnabrückGermany
  2. 2.Fachbereich MathematikUniversität SalzburgSalzburgAustria

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