The Journal of Geometric Analysis

, Volume 28, Issue 1, pp 405–426 | Cite as

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

  • Julia Hörrmann
  • Joscha Prochno
  • Christoph Thäle


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.


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

Mathematics Subject Classification

52A20 52B11 60D05 



We would like to thank David Alonso-Gutiérrez and Apostolos Giannopoulos for useful conversations and interesting hints and remarks. We would also like to thank an anonymous referee for many helpful suggestions and especially for pointing us to an error in an earlier version of this manuscript. The financial support of the Mercator Research Center Ruhr has made possible a research stay of the second author at Ruhr University Bochum.


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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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