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

The Gaussian Isoperimetric Inequality and Transportation

  • Gordon Blower


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 


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  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. ( 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

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© Kluwer Academic Publishers 2003

Authors and Affiliations

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

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