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 ThäleEmail author


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.


  1. 1.
    Alonso-Gutiérrez, D.: On the isotropy constant of random convex sets. Proc. Am. Math. Soc. 136(9), 3293–3300 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alonso-Gutiérrez, D.: A remark on the isotropy constant of polytopes. Proc. Am. Math. Soc. 139(7), 2565–2569 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alonso-Gutirrez, D., Bastero, J., Bernus, J., Wolff, P.: On the isotropy constant of projections of polytopes. J. Funct. Anal. 258(5), 1452–1465 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alonso-Gutiérrez, D., Litvak, A.E., Tomczak-Jaegermann, N.: On the isotropic constant of random polytopes. J. Geom. Anal. 26(1), 645–662 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Artstein-Avidan, S., Giannopoulos, A., Milman, V.D.: Asymptotic geometric analysis. Part I, volume 202 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2015)Google Scholar
  6. 6.
    Ball, K.: Normed spaces with a weak-Gordon-Lewis property. In: Functional analysis (Austin, TX, 1987/1989), volume 1470 of Lecture Notes in Mathematics, pp. 36–47. Springer, Berlin, (1991)Google Scholar
  7. 7.
    Barthe, F., Guédon, O., Mendelson, S., Naor, A.: A probabilistic approach to the geometry of the \(l^n_p\)-ball. Ann. Probab. 33(2), 480–513 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bobkov, S.G., Nazarov, F.L.: Large deviations of typical linear functionals on a convex body with unconditional basis. In: Stochastic inequalities and applications, volume 56 of Progress in Probability, pp. 3–13. Birkhäuser, Basel (2003)Google Scholar
  9. 9.
    Böröczky, K.J., Fodor, F., Hug, D.: Intrinsic volumes of random polytopes with vertices on the boundary of a convex body. Trans. Am. Math. Soc. 365(2), 785–809 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bourgain, J.: On high-dimensional maximal functions associated to convex bodies. Am. J. Math. 108(6), 1467–1476 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bourgain, J.: On the distribution of polynomials on high dimensional convex sets. In: Geometric aspects of functional analysis, volume 1469 of Lecture Notes in Mathematics, pp. 127–137. Springer, Berlin (1991)Google Scholar
  12. 12.
    Bourgain, J., Lindenstrauss, J., Milman, V.D.: Minkowski sums and symmetrizations. In: Geometric aspects of functional analysis, volume 1317 of Lecture Notes in Mathematics, pp. 44–74. Springer, Berlin (1988)Google Scholar
  13. 13.
    Brazitikos, S., Giannopoulos, A., Valettas, P., Vritsiou, B.-H.: Geometry of Isotropic Convex Bodies. Mathematical Surveys and Monographs, vol. 196. American Mathematical Society, Providence, RI (2014)zbMATHGoogle Scholar
  14. 14.
    Dafnis, N., Giannopoulos, A., Tsolomitis, A.: Asymptotic shape of a random polytope in a convex body. J. Funct. Anal. 257(9), 2820–2839 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dafnis, N., Giannopoulos, A., Guédon, O.: On the isotropic constant of random polytopes. Adv. Geom. 10(2), 311–322 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Giannopoulos, A., Hioni, L., Tsolomitis, A.: Asymptotic shape of the convex hull of isotropic log-concave random vectors. Adv. Appl. Math. 75, 116–143 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gluskin, E.D.: The diameter of the Minkowski compactum is roughly equal to \(n\). Funktsional. Anal. Prilozhen. 15(1), 72–73 (1981)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Gluskin, E.D., Kwapień, S.: Tail and moment estimates for sums of independent random variables with logarithmically concave tails. Stud. Math. 114(3), 303–309 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hensley, D.: Slicing convex bodies–bounds for slice area in terms of the body’s covariance. Proc. Am. Math. Soc. 79(4), 619–625 (1980)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hörrmann, J., Hug, D.: On the volume of the zero cell of a class of isotropic Poisson hyperplane tessellations. Adv. Appl. Probab. 46, 622–642 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hörrmann, J., Hug, D., Reitzner, M., Thäle, C.: Poisson polyhedra in high dimensions. Adv. Math. 281, 1–39 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Junge, M.: Hyperplane conjecture for quotient spaces of lp. Forum Math. 6(5), 617–636 (1994)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Klartag, B.: On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal. 16(6), 1274–1290 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Klartag, B., Kozma, G.: On the hyperplane conjecture for random convex sets. Israel J. Math. 170, 253–268 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Klartag, B., Milman, E.: Centroid bodies and the logarithmic Laplace transform—a unified approach. J. Funct. Anal. 262(1), 10–34 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    König, H., Meyer, M., Pajor, A.: The isotropy constants of the Schatten classes are bounded. Math. Ann. 312(4), 773–783 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Milman, V.D., Pajor, A.: Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed \(n\)-dimensional space. In: Geometric aspects of functional analysis (1987–1988), volume 1376 of Lecture Notes in Mathematics, pp. 64–104. Springer, Berlin (1989)Google Scholar
  28. 28.
    Naor, A.: The surface measure and cone measure on the sphere of \(\ell _p^n\). Trans. Am. Math. Soc. 359(3), 1045–1079 (2007)CrossRefzbMATHGoogle Scholar
  29. 29.
    Naor, A., Romik, D.: Projecting the surface measure of the sphere of \(\ell _p^n\). Ann. Inst. H. Poincaré Probab. Stat. 39(2), 241–261 (2003)CrossRefzbMATHGoogle Scholar
  30. 30.
    Paouris, G.: Concentration of mass and central limit properties of isotropic convex bodies. Proc. Am. Math. Soc. 133(2), 565–575 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Paouris, G.: On the \(\psi _2 \)-behaviour of linear functionals on isotropic convex bodies. Stud. Math. 168, 285–299 (2005)CrossRefzbMATHGoogle Scholar
  32. 32.
    Paouris, G.: Concentration of mass on convex bodies. Geom. Funct. Anal. 16(5), 1021–1049 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Pivovarov, P.: On determinants and the volume of random polytopes in isotropic convex bodies. Geom. Dedicata 149(1), 45–58 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Rachev, S.T., Rüschendorf, L.: Approximate independence of distributions on spheres and their stability properties. Ann. Probab. 19(3), 1311–1337 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Reitzner, M.: Random points on the boundary of smooth convex bodies. Trans. Am. Math. Soc. 354(6), 2243–2278 (2002). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Reitzner, M.: Random polytopes and the Efron–Stein jackknife inequality. Ann. Probab. 31(4), 2136–2166 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Richardson, R.M., Vu, V.H., Wu, L.: An inscribing model for random polytopes. Discret. Comput. Geom. 39(1–3), 469–499 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Schechtman, G., Zinn, J.: On the volume of the intersection of two \(L^n_p\) balls. Proc. Am. Math. Soc. 110(1), 217–224 (1990)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Schechtman, G., Zinn, J.: Geometric Aspects of Functional Analysis: Israel Seminar 1996–2000, Chapter Concentration on the \(\ell _p^n\) ball, pages 245–256. Springer, Berlin (2000)Google Scholar
  40. 40.
    Schütt, C., Werner, E.: Polytopes with vertices chosen randomly from the boundary of a convex body. In: Geometric aspects of functional analysis, volume 1807 of Lecture Notes in Mathematics, pp. 241–422. Springer, Berlin (2003)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

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

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