Random Approximation of Convex Bodies: Monotonicity of the Volumes of Random Tetrahedra View Full Text


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Article Info

DATE

2017-08-07

AUTHORS

Stefan Kunis, Benjamin Reichenwallner, Matthias Reitzner

ABSTRACT

Choose uniform random points X1,⋯,Xn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1, \dots , X_n$$\end{document} in a given convex set and let conv[X1,⋯,Xn]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text { conv}}[X_1, \dots , X_n]$$\end{document} 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. More... »

PAGES

165-174

References to SciGraph publications

  • 2003-08. The Expected Volume of a Tetrahedron whose Vertices are Chosen at Random in the Interior of a Cube in MONATSHEFTE FÜR MATHEMATIK
  • 2012-10-13. Random Polytopes in STOCHASTIC GEOMETRY, SPATIAL STATISTICS AND RANDOM FIELDS
  • 2006-01-01. Disciplined Convex Programming in GLOBAL OPTIMIZATION
  • 2008. Stochastic and Integral Geometry in NONE
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00454-017-9914-7

    DOI

    http://dx.doi.org/10.1007/s00454-017-9914-7

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    https://app.dimensions.ai/details/publication/pub.1091082566


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