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 


  1. 1.
    Alonso-Gutiérrez, D.: On the isotropy constant of random convex sets. Proc. Am. Math. Soc. 136(9), 3293–3300 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alonso-Gutiérrez, D.: A remark on the isotropy constant of polytopes. Proc. Am. Math. Soc. 139(7), 2565–2569 (2011)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bourgain, J.: On high-dimensional maximal functions associated to convex bodies. Am. J. Math. 108(6), 1467–1476 (1986)MathSciNetCrossRefMATHGoogle 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)MATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dafnis, N., Giannopoulos, A., Guédon, O.: On the isotropic constant of random polytopes. Adv. Geom. 10(2), 311–322 (2010)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hörrmann, J., Hug, D., Reitzner, M., Thäle, C.: Poisson polyhedra in high dimensions. Adv. Math. 281, 1–39 (2015)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Junge, M.: Hyperplane conjecture for quotient spaces of lp. Forum Math. 6(5), 617–636 (1994)MathSciNetMATHGoogle Scholar
  23. 23.
    Klartag, B.: On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal. 16(6), 1274–1290 (2006)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Klartag, B., Kozma, G.: On the hyperplane conjecture for random convex sets. Israel J. Math. 170, 253–268 (2009)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Paouris, G.: On the \(\psi _2 \)-behaviour of linear functionals on isotropic convex bodies. Stud. Math. 168, 285–299 (2005)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Paouris, G.: Concentration of mass on convex bodies. Geom. Funct. Anal. 16(5), 1021–1049 (2006)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Pivovarov, P.: On determinants and the volume of random polytopes in isotropic convex bodies. Geom. Dedicata 149(1), 45–58 (2010)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Reitzner, M.: Random points on the boundary of smooth convex bodies. Trans. Am. Math. Soc. 354(6), 2243–2278 (2002). (electronic)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Reitzner, M.: Random polytopes and the Efron–Stein jackknife inequality. Ann. Probab. 31(4), 2136–2166 (2003)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle 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
  1. 1.Faculty of MathematicsRuhr University BochumBochumGermany
  2. 2.School of Mathematics & Physical SciencesUniversity of HullHullUK

Personalised recommendations