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Perturbations in the Gaussian isoperimetric inequality

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An isoperimetric inequality of Gaussian type is derived for the class of probability measures on the Euclidean space, having perturbed log-concave densities with respect to the standard Gaussian measure. Bibliography: 23 titles.

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References

  1. V. N. Sudakov, B. S. Cirel’son (B. S. Tsirel’son), “Extremal properties of half-spaces for spherically invariant measures” [in Russian], Zap. Nauchn. Semin. LOMI 41, 14–24 (1974); English transl.: J. Sov. Math. 9, 9–18 (1978).

    MATH  Google Scholar 

  2. C. Borell, “The Brunn-Minkowski inequality in Gauss space,” Invent. Math. 30, No. 2, 207–216 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Ehrhard, “Symétrisation dans l’espace de Gauss,” Math. Scand. 53, 281–301 (1983).

    MATH  MathSciNet  Google Scholar 

  4. Yu. A. Davydov, M. A. Lifshits, N. V. Smorodina, Local Properties of Distributions of Stochastic Functionals, Am. Math. Soc., Providence, RI (1998).

    MATH  Google Scholar 

  5. R. Latala, “A note on the Ehrhard inequality,” Studia Math. 118, No. 2, 169–174 (1996).

    MATH  MathSciNet  Google Scholar 

  6. C. Borell, “The Ehrhard inequality,” C. R. Math. Acad. Sci. Paris 337, No. 10, 663–666 (2003).

    MATH  MathSciNet  Google Scholar 

  7. C. Borell, “Inequalities of the Brunn-Minkowski type for Gaussian measures,” Probab. Theory Related Fields 140, No. 1–2, 195–205 (2008).

    MATH  MathSciNet  Google Scholar 

  8. D. Bakry, M. Ledoux, “Lévy-Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator,” Invent. Math. 123, No. 2, 259–281 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Ledoux, “A short proof of the Gaussian isoperimetric inequality. High dimensional probability” In: Oberwolfach, 1996, pp. 229–232, Progr. Probab. 43, Birkhäuser, Basel (1998).

    Google Scholar 

  10. S. G. Bobkov, “An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space,” Ann. Probab. 25, No. 1, 206–214 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  11. S. G. Bobkov, “A localized proof of the isoperimetric Bakry-Ledoux inequality and some applications” [in Russian], Teor. Veroyatnost. Primenen. 47, No. 2, 340–346 (2002); English transl.: Theory Probab. Appl. 47, No. 2, 308–314 (2003).

    MathSciNet  Google Scholar 

  12. L. A. Caffarelli, “Monotonicity properties of optimal transportation and the FKG and related inequalities,” Commun. Math. Phys. 214, 547–563 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  13. S. G. Bobkov, “A functional form of the isoperimetric inequality for the Gaussian measure,” J. Funct. Anal. 135, No. 1, 39–49 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Ledoux, “Concentration of measure and logarithmic Sobolev inequalities,” In: Séminaire de Probabilités. Lect. Notes Math. 1709 120–216 (1999).

    MathSciNet  Google Scholar 

  15. L. Gross, “Logarithmic Sobolev inequalities.” Am. J. Math. 97, 1061–1083 (1975).

    Article  Google Scholar 

  16. L. Lovász, M. Simonovits, “Random walks in a convex body and an improved volume algorithm,” Random Struct. Algorithms 4, No. 3, 359–412 (1993).

    Article  MATH  Google Scholar 

  17. M. Fradelizi, O. Guédon, “The extreme points of subsets of s-concave probabilities and a geometric localization theorem,” Discrete Comput. Geom. 31, No. 2, 327–335 (2004).

    MATH  MathSciNet  Google Scholar 

  18. M. Fradelizi, O. Guédon, “A generalized localization theorem and geometric inequalities for convex bodies,” Adv. Math. 204, No. 2, 509–529 (2006),

    Article  MATH  MathSciNet  Google Scholar 

  19. R. Kannan, L. Lovász, M. Simonovits, “Isoperimetric problems for convex bodies and a localization lemma,” Discrete Comput. Geom. 13, 541–559 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  20. S. G. Bobkov, C. Houdré, “Isoperimetric constants for product probability measures,” Ann. Probab. 25, No. 1, 184–205 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  21. F. Barthe, P. Cattiaux, C. Roberto, “Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry,” Rev. Mat. Iberoam. 22, No. 3, 993–1067 (2006).

    MATH  MathSciNet  Google Scholar 

  22. F. Barthe, P. Cattiaux, C. Roberto, “Isoperimetry between exponential and Gaussian,” Electron. J. Probab. 12, No. 44, 1212–1237 (2007).

    MATH  MathSciNet  Google Scholar 

  23. S. G. Bobkov, “On the isoperimetric constants for product measures” [in Russian], Probl. Mat. Anal. 40, 49–56 (2009); English transl.: J. Math. Sci., New York 159, No. 1, 47–53 (2009).

    Google Scholar 

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  1. University of Minnesota, 127 Vincent Hall, Minneapolis, MN, 55455, USA

    S. G. Bobkov

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Correspondence to S. G. Bobkov.

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Translated from Problems in Mathematical Analysis 45, February 2010, pp. 3–14.

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Bobkov, S.G. Perturbations in the Gaussian isoperimetric inequality. J Math Sci 166, 225–238 (2010). https://doi.org/10.1007/s10958-010-9864-6

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  • DOI: https://doi.org/10.1007/s10958-010-9864-6

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