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Poincaré inequality in mean value for Gaussian polytopes
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  • Published: 22 September 2010

Poincaré inequality in mean value for Gaussian polytopes

  • B. Fleury1 

Probability Theory and Related Fields volume 152, pages 141–178 (2012)Cite this article

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Abstract

Let K N = [±G 1, . . . , ±G N ] be the absolute convex hull of N independent standard Gaussian random points in \({\mathbb R^n}\) with N ≥ n. We prove that, for any 1-Lipschitz function \({f:\mathbb R^n\rightarrow\mathbb R}\), the polytope K N satisfies the following Poincaré inequality in mean value:

$$\mathbb {E}_{\omega} \int\limits_{K_N(\omega)} \left( f(x) - \frac{1}{\textup{vol}_n\left(K_N(\omega)\right)} \int\limits_{K_n(\omega)}f(y)dy \right)^2 dx \leq \frac{C}{n} \mathbb E_{\omega} \int\limits_{K_N(\omega)}|x|^2dx$$

where C > 0 is an absolute constant. This Poincaré inequality is the one suggested by a conjecture of Kannan, Lovász and Simonovits for general convex bodies. Moreover, we prove in mean value that the volume of the polytope K N is concentrated in a subexponential way within a thin Euclidean shell with the optimal dependence of the dimension n. An important tool of the proofs is a representation of the law of (G 1, . . . , G n ) conditioned by the event that “the convex hull of G 1, . . . , G n is a (n − 1)-face of K N ”. As an application, we also get an estimate of the number of (n − 1)-faces of the polytope K N , valid for every N ≥ n.

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References

  1. Affentranger F., Schneider R.: Random projections of regular simplices. Discrete Comput. Geom. 7 219–226 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anttila M., Ball K., Perissinaki I.: The central limit problem for convex bodies. Trans. Am. Math. Soc. 355(12), 4723–4735 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bakry, D., Emery, M.: Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/84. Lecture Notes in Mathematics, vol. 1123, pp. 177–206. Springer, Berlin (1985)

  4. Ball K.: Logarithmically concave functions and sections of convex sets. Stud. Math. 88, 69–84 (1988)

    MATH  Google Scholar 

  5. Barthe F., Wolff P.: Remarks on non-interacting conservative spin systems: the case of gamma distributions. Stoch. Proc. Appl. 119, 2711–2723 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baryshnikov Y.M., Vitale R.A.: Regular simplices and Gaussian samples. Discrete Comput. Geom. 11, 141–147 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bobkov S.G.: Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27(4), 1903–1921 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bobkov, S.G.: Remarks on the growth of l p-norms of polynomials. In: Geometric Aspects of Functionnal Analysis. Lecture Notes in Mathemtics, vol. 1745, pp. 27–35. Springer, Berlin (2000)

  9. Bobkov, S.G.: On isoperimetric constants for log-concave probability distributions. In: Geometric Aspects of Functional Analysis. Israel Seminar 2004–2005, Lecture Notes in Mathematics, vol. 1910, pp. 81–88. Springer, Berlin (2007)

  10. Borell C.: Convex measures on locally convex spaces. Ark. Mat. 12, 239–252 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bourgain, J.: On the distribution of polynomials on high dimensional convex sets. In: Israel Seminar (GAFA) 1989–1990. Lecture Notes in Mathemtics, vol. 1469, pp. 127–137 (1991)

  12. Buser P.: A note on the isoperimetric constant. Ann. Sci. Ecole Norm. Sup. (4) 15(2), 213–230 (1982)

    MathSciNet  MATH  Google Scholar 

  13. Carl B., Pajor A.: Gelfand numbers of operators with values in Hilbert space. Invent. Math. 94, 479–504 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis (Paper dedicated to Salomon Bochner, 1969), pp. 195–199. Princeton University Press, Princeton (1970)

  15. Dafnis N., Giannopoulos A., Guédon O.: On the isotropic constant of random polytopes. Adv. Geom. 10, 311–321 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dyer, M., Frieze, A.: Computing the volume of convex bodies: a case where randomness provably helps. In: Probabilistic Combinatorics and its Applications (ed. Béla Bollobás). Proceedings of Symposia in Applied Mathematics, vol. 44, pp. 123–170 (1992)

  17. Fleury B.: Concentration in a thin Euclidean shell for log-concave measures. J. Funct. Anal. 259(4), 832–841 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fleury B., Guédon O., Paouris G.: A stability result for mean width of L p -centroid bodies. Adv. Math. 214(2), 865–877 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gluskin E.D.: The diameter of Minkowski compactum roughly equals to n. Funct. Anal. Appl. 15, 57–58 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gluskin E.D.: Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces. Translation in Math. USSR-Sb. 64(1), 85–96 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gromov M., Milman V.D.: A topological application of the isoperimetric inequality. Am. J. Math. 105(4), 843–854 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hsu, E.: Stochastic analysis on manifolds. In: Graduate Studies in Mathematics, vol. 38. American Mathematical Society (2002)

  23. Huet, N.: Inégalités géométriques pour des mesures log-concaves. PhD thesis, Institut de MathTmatiques de Toulouse (2009)

  24. Hug D., Munsonius G.O., Reitzner M.: Asymptotic mean values of Gaussian polytopes. Beitr. Algebra Geom. 45, 531–548 (2004)

    MathSciNet  MATH  Google Scholar 

  25. Hug D., Reitzner M.: Gaussian polytopes: variances and limit theorems. Adv. Appl. Prob. 37, 297–320 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kannan R., Lovász L., Simonovits M.: Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13(3–4), 541–559 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  27. Karzanov A., Khachiyan L.G.: On the conductance of order Markov chains. Order 8, 7–15 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. Klartag B.: A central limit theorem for convex sets. Invent. Math. 168, 91–131 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Klartag B.: Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245, 284–310 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Klartag B.: A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Relat. Fields 45(1), 1–33 (2009)

    Article  MathSciNet  Google Scholar 

  31. Klartag B., Kozma G.: On the hyperplane conjecture for random convex sets. Isr. J. Math. 170(1), 253–268 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Latala R., Wojtaszczyk J.O.: On the infimum convolution inequality. Stud. Math. 189(3), 147–187 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ledoux, M.: The Concentration of Measure Phenomenon. American Mathematical Society, Providence (2001)

  34. Ledoux, M.: Spectral gap, logarithmic Sobolev constant, and geometric bounds. In: Surveys in Differential Geometry, vol. IX, pp. 219–240. Int. Press, Somerville (2004)

  35. Litvak A.E., Pajor A., Rudelson M., Tomczak-Jaegermann N.: Smallest singular value of random matrices and geometry of random polytopes. Adv. Math. 195, 491–523 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lovász, L., Simonovits, M.: Mixing rate of Markov chains, an isoperimetric inequality, and computing the volume. In: Proceedings of 31st Annual Symposium on Found. of Computer Science, pp. 346–355. IEEE Computer Society (1990)

  37. Maz’ja V.G.: The negative spectrum of the higher-dimensional Schrödinger operator. Dokl. Akad. Nauk. SSSR 144, 721–722 (1962)

    MathSciNet  Google Scholar 

  38. Maz’ja V.G.: On the solvability of the Neumann problem. Dokl. Akad. Nauk. SSSR 147, 294–296 (1962)

    MathSciNet  Google Scholar 

  39. Milman E.: On the role of convexity in isoperimetry, spectral-gap and concentration. Invent. Math. 177(1), 1–43 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Nazarov, F.: On the maximal perimeter of a convex set in \({\mathbb{R}^n}\) with respect to a Gaussian measure. In: Geometric Aspects of Functional Analysis (2001–2002). Lecture Notes in Mathematics, vol. 1807, pp. 169–187. Springer, Berlin (2003)

  41. Payne L.E., Weinberger H.F.: An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5, 286–292 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  42. Raynaud H.: Sur l’enveloppe convexe des nuages de points aléatoires dans \({\mathbb{R}^n}\) I. J. Appl. Probab. 7, 35–48 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  43. Schneider R., Weil W.: Integralgeometrie. Teubner, Stuttgart (1992)

    MATH  Google Scholar 

  44. Sodin S.: An isoperimetric inequality on the l p balls. Ann. Inst. H.Poincaré Probab. Stat. 44(2), 362–373 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  45. Szarek S.J.: The finite-dimensional basis problem with an appendix on nets of Grassman manifold. Acta Math. 141, 153–179 (1983)

    Article  MathSciNet  Google Scholar 

  46. Van der Vaart A.W., Wellner J.A.: Weak Convergence and Emperical Processes. Springer-Verlag, Berlin (1996)

    Google Scholar 

  47. Wojtaszczyk, J.O.: The square negative correlation property for generalized Orlicz balls. In: Geometric Aspects of Functional Analysis—Israel Seminar. Lecture Notes in Mathematics, vol. 1910, pp. 305–313. Springer, Berlin (2004)

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  1. Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, boîte 186, 4 Place Jussieu, 75252, Paris Cedex 05, France

    B. Fleury

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Fleury, B. Poincaré inequality in mean value for Gaussian polytopes. Probab. Theory Relat. Fields 152, 141–178 (2012). https://doi.org/10.1007/s00440-010-0318-3

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  • Received: 29 September 2009

  • Revised: 20 August 2010

  • Published: 22 September 2010

  • Issue Date: February 2012

  • DOI: https://doi.org/10.1007/s00440-010-0318-3

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Mathematics Subject Classification (2000)

  • 52B11
  • 52B60
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