Advertisement

Positivity

, Volume 7, Issue 3, pp 203–224 | Cite as

The Gaussian Isoperimetric Inequality and Transportation

  • Gordon Blower
Article

Abstract

Any probability measure on \(\mathbb{R}\)d which satisfies the Gaussian or exponential isoperimetric inequality fulfils a transportation inequality for a suitable cost function. Suppose that W (x) dx satisfies the Gaussian isoperimetric inequality: then a probability density function f with respect to W (x) dx has finite entropy, provided that ∈t\(\smallint \left\| {\nabla {\text{ }}f} \right\|\{ \log _{\text{ + }} \left\| {\nabla {\text{ }}f} \right\|\} ^{1/2} {\text{ }}W(x){\text{ d}}x < \infty \). This strengthens the quadratic logarithmic Sobolev inequality of Gross (Amr. J. Math 97 (1975) 1061). Let μ(dx) = e−ξ(x) dx be a probability measure on \(\mathbb{R}\)d, where ξ is uniformly convex. Talagrand's technique extends to monotone rearrangements in several dimensions (Talagrand, Geometric Funct. Anal. 6 (1996) 587), yielding a direct proof that μ satisfies a quadratic transportation inequality. The class of probability measures that satisfy a quadratic transportation inequality is stable under multiplication by logarithmically bounded Lipschitz densities.

isoperimetric function transportation logarithmic Sobolev inequality Orlicz spaces 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bakry, D. and Ledoux, M.: Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator, Invent. Math. 123 (1996), 259–281.Google Scholar
  2. 2.
    Bobkov, S.G.: Isoperimetric and analytic inequalities for log-concave probability measures, Annals Probab. 27 (1999), 1903–1921.Google Scholar
  3. 3.
    Bobkov, S.G. and Götze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities,J. Funct. Anal. 162 (1999), 1–28.Google Scholar
  4. 4.
    Bobkov, S.G., Gentil I. and Ledoux, M.: Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl. (9) 80 (2001), 669–696.Google Scholar
  5. 5.
    Bobkov, S.G. and Ledoux, M.: From Brunn-Minkowski to Brescamp-Lieb and to logarithmic Sobolev inequalities, Geometric and Funct. Anal. 10 (2000), 1028–1052.Google Scholar
  6. 6.
    Borell, C.: The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), 207–216.Google Scholar
  7. 7.
    Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure. Appl. Math. 44 (1991), 375–417.Google Scholar
  8. 8.
    Caffarelli, L.A.: The regularity of mappings with a convex potential, J. Amr. Math. Soc. 5 (1992), 99–104.Google Scholar
  9. 9.
    Carlen, E.A. and Carvallo, M.C.: Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation, J. Statist. Phys. 67 (1992), 575–608.Google Scholar
  10. 10.
    Chavel, I.: Riemannian Geometry: A Modern Introduction, Cambridge University Press, Cambridge, 1993.Google Scholar
  11. 11.
    Deuschel, J.-D. and Stroock, D.W.:Hypercontractivity and spectral gap of symmetric diffusions with applications to the stochastic Ising models, J. Funct. Anal. 92 (1990), 30–48.Google Scholar
  12. 12.
    Dudley, R.M.: Real Analysis and Probability, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1989.Google Scholar
  13. 13.
    Federer, H.: Geometric Measure Theory, Springer, Berlin, 1969.Google Scholar
  14. 14.
    Gangbo, W. and McCann, R.J.: The geometry of optimal transportation, Acta Math. 177 (1996), 113–161.Google Scholar
  15. 15.
    Gromov, M.: Isoperimetric inequalities in Riemannian manifolds, Appendix pp. 114–129; in V.D. Milman and G. Schechtman, (eds) Asymptotic Theory of Finite Dimensional Normed Spaces, Springer Lecture Notes in Mathematics 1200, Springer, Berlin, 1986.Google Scholar
  16. 16.
    Gross, L.: Logarithmic Sobolev inequalities, Amr. J. Math. 97 (1975), 1061–1083.Google Scholar
  17. 17.
    Ledoux, M.: Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space,Bull. Sci. Math. 118 (1994), 485–510.Google Scholar
  18. 18.
    Ledoux, M.: A simple analytic proof of an inequality by P. Buser, Proc. Amr. Math. Soc. 121 (1994), 951–959.Google Scholar
  19. 19.
    Ledoux, M.: Isopérimétrie et inégalités de Sobolev logarithmiques gaussiennes, C.R. Acad. Sci. Math. 306 (1988), 79–82.Google Scholar
  20. 20.
    Ledoux, M.: Concentration of Measure and Logarithmic Sobolev Inequalities, pp. 120–216 in Séminaire de Probabilités XXXIII. Lecture Notes in Math., 1709 Springer, Berlin 1999. (http://www-sv.cict.fr/lsp./Ledoux/).Google Scholar
  21. 21.
    Lindenstrauss, J. and Tzafriri, L.: Classical Banach Spaces I and II, Springer, New York, 1977.Google Scholar
  22. 22.
    Maz'ja, V.G.: Sobolev spaces, Springer, Berlin, 1985.Google Scholar
  23. 23.
    McCann, R.J.: A convexity principle for interacting gases, Adv. Math. 128 (1997), 153–179.Google Scholar
  24. 24.
    McCann, R.J.: Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80 (1995), 309–323.Google Scholar
  25. 25.
    Otto, F. and Villani, C.: Generalizations of an inequality by Talagrand, and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000), 361–400.Google Scholar
  26. 26.
    Pelczy´nski, A. and Wojciechowski, M.: Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm, Studia Math. 107 (1993), 61–100.Google Scholar
  27. 27.
    Sudakov, V.N. and Tsirel'son, B.S.: Extremal properties of half-spaces for spherically invariant measures, J. Soviet Math. 9 (1978), 9–18.Google Scholar
  28. 28.
    Talagrand, M.: Transportation cost for gaussian and other product measures, Geometric and Funct. Anal. 6 (1996), 587–600.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Gordon Blower
    • 1
  1. 1.Department of Mathematics and StatisticsLancaster UniversityLancasterUK

Personalised recommendations