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

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.

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Acknowledgements

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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Correspondence to Christoph Thäle.

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Hörrmann, J., Prochno, J. & Thäle, C. On the Isotropic Constant of Random Polytopes with Vertices on an \(\ell _p\)-Sphere. J Geom Anal 28, 405–426 (2018). https://doi.org/10.1007/s12220-017-9826-z

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Keywords

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

Mathematics Subject Classification

  • 52A20
  • 52B11
  • 60D05