On the Isotropic Constant of Random Polytopes with Vertices on an \(\ell _p\)-Sphere

  • Julia Hörrmann
  • Joscha Prochno
  • Christoph Thäle
Article

Abstract

The symmetric convex hull of random points that are independent and distributed according to the cone probability measure on the \(\ell _p\)-unit sphere of \({{\mathbb {R}}}^n\) for some \(1\le p < \infty \) is considered. We prove that these random polytopes have uniformly absolutely bounded isotropic constants with overwhelming probability. This generalizes the result for the Euclidean sphere \((p=2)\) obtained by Alonso-Gutiérrez. The proof requires several different tools including a probabilistic representation of the cone measure due to Schechtman and Zinn and moment estimates for sums of independent random variables with log-concave tails originating in a paper of Gluskin and Kwapień.

Keywords

Asymptotic convex geometry Cone measure Hyperplane conjecture Isotropic constant \(\ell _p\)-Sphere Random polytope Stochastic geometry 

Mathematics Subject Classification

52A20 52B11 60D05 

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Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  • Julia Hörrmann
    • 1
  • Joscha Prochno
    • 2
  • Christoph Thäle
    • 1
  1. 1.Faculty of MathematicsRuhr University BochumBochumGermany
  2. 2.School of Mathematics & Physical SciencesUniversity of HullHullUK

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