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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Alonso-Gutiérrez, D.: On the isotropy constant of random convex sets. Proc. Am. Math. Soc. 136(9), 3293–3300 (2008)
Alonso-Gutiérrez, D.: A remark on the isotropy constant of polytopes. Proc. Am. Math. Soc. 139(7), 2565–2569 (2011)
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)
Alonso-Gutiérrez, D., Litvak, A.E., Tomczak-Jaegermann, N.: On the isotropic constant of random polytopes. J. Geom. Anal. 26(1), 645–662 (2016)
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)
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)
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)
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)
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)
Bourgain, J.: On high-dimensional maximal functions associated to convex bodies. Am. J. Math. 108(6), 1467–1476 (1986)
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)
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)
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)
Dafnis, N., Giannopoulos, A., Tsolomitis, A.: Asymptotic shape of a random polytope in a convex body. J. Funct. Anal. 257(9), 2820–2839 (2009)
Dafnis, N., Giannopoulos, A., Guédon, O.: On the isotropic constant of random polytopes. Adv. Geom. 10(2), 311–322 (2010)
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)
Gluskin, E.D.: The diameter of the Minkowski compactum is roughly equal to \(n\). Funktsional. Anal. Prilozhen. 15(1), 72–73 (1981)
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)
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)
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)
Hörrmann, J., Hug, D., Reitzner, M., Thäle, C.: Poisson polyhedra in high dimensions. Adv. Math. 281, 1–39 (2015)
Junge, M.: Hyperplane conjecture for quotient spaces of lp. Forum Math. 6(5), 617–636 (1994)
Klartag, B.: On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal. 16(6), 1274–1290 (2006)
Klartag, B., Kozma, G.: On the hyperplane conjecture for random convex sets. Israel J. Math. 170, 253–268 (2009)
Klartag, B., Milman, E.: Centroid bodies and the logarithmic Laplace transform—a unified approach. J. Funct. Anal. 262(1), 10–34 (2012)
König, H., Meyer, M., Pajor, A.: The isotropy constants of the Schatten classes are bounded. Math. Ann. 312(4), 773–783 (1998)
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)
Naor, A.: The surface measure and cone measure on the sphere of \(\ell _p^n\). Trans. Am. Math. Soc. 359(3), 1045–1079 (2007)
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)
Paouris, G.: Concentration of mass and central limit properties of isotropic convex bodies. Proc. Am. Math. Soc. 133(2), 565–575 (2005)
Paouris, G.: On the \(\psi _2 \)-behaviour of linear functionals on isotropic convex bodies. Stud. Math. 168, 285–299 (2005)
Paouris, G.: Concentration of mass on convex bodies. Geom. Funct. Anal. 16(5), 1021–1049 (2006)
Pivovarov, P.: On determinants and the volume of random polytopes in isotropic convex bodies. Geom. Dedicata 149(1), 45–58 (2010)
Rachev, S.T., Rüschendorf, L.: Approximate independence of distributions on spheres and their stability properties. Ann. Probab. 19(3), 1311–1337 (1991)
Reitzner, M.: Random points on the boundary of smooth convex bodies. Trans. Am. Math. Soc. 354(6), 2243–2278 (2002). (electronic)
Reitzner, M.: Random polytopes and the Efron–Stein jackknife inequality. Ann. Probab. 31(4), 2136–2166 (2003)
Richardson, R.M., Vu, V.H., Wu, L.: An inscribing model for random polytopes. Discret. Comput. Geom. 39(1–3), 469–499 (2008)
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)
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)
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)
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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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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DOI: https://doi.org/10.1007/s12220-017-9826-z
Keywords
- Asymptotic convex geometry
- Cone measure
- Hyperplane conjecture
- Isotropic constant
- \(\ell _p\)-Sphere
- Random polytope
- Stochastic geometry