Advertisement

Rearrangement and Prékopa–Leindler Type Inequalities

  • James MelbourneEmail author
Conference paper
Part of the Progress in Probability book series (PRPR, volume 74)

Abstract

We investigate the interactions of functional rearrangements with Prékopa–Leindler type inequalities. It is shown that certain set theoretic rearrangement inequalities can be lifted to functional analogs, thus demonstrating that several important integral inequalities tighten on functional rearrangement about “isoperimetric” sets with respect to a relevant measure. Applications to the Borell–Brascamp–Lieb, Borell–Ehrhard, and the recent polar Prékopa–Leindler inequalities are demonstrated. It is also proven that an integrated form of the Gaussian log-Sobolev inequality sharpens on rearrangement.

References

  1. 1.
    S. Artstein-Avidan, D. Florentin, A. Segal, Polar Prékopa–Leindler inequalities (2017). arXiv preprint. arXiv:1707.08732Google Scholar
  2. 2.
    F. Barthe, On a reverse form of the Brascamp-Lieb inequality. Invent. Math. 134(2), 335–361 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    F. Barthe, N. Huet, On Gaussian Brunn–Minkowski inequalities. Stud. Math. 191(3), 283–304 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    W. Beckner, Inequalities in Fourier analysis. Ann. Math. 102(1), 159–182 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    N.M. Blachman, The convolution inequality for entropy powers. IEEE Trans. Inf. Theory 11, 267–271 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    S.G. Bobkov, G.P. Chistyakov, Bounds for the maximum of the density of the sum of independent random variables. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 408, 62–73, 324 (2012)Google Scholar
  7. 7.
    S.G. Bobkov, G.P. Chistyakov, Entropy power inequality for the Rényi entropy. IEEE Trans. Inf. Theory 61(2), 708–714 (2015)zbMATHCrossRefGoogle Scholar
  8. 8.
    S.G. Bobkov, M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10(5), 1028–1052 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    S.G. Bobkov, M. Ledoux, Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. 37(2), 403–427 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    S. Bobkov, A. Marsiglietti, Variants of the entropy power inequality. IEEE Trans. Inf. Theory 63(12), 7747–7752 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    S. Bobkov, J. Melbourne, Hyperbolic measures on infinite dimensional spaces. Probab. Surv. 13, 57–88 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    S.G. Bobkov, I. Gentil, M. Ledoux, Hypercontractivity of Hamilton–Jacobi equations. J. Math. Pures Appl. 80(7), 669–696 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    C. Borell, Convex measures on locally convex spaces. Ark. Mat. 12, 239–252 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    C. Borell, The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30(2), 207–216 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    C. Borell, Convex set functions in d-space. Period. Math. Hungar. 6(2), 111–136 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    C. Borell, Inequalities of the Brunn-Minkowski type for Gaussian measures. Probab. Theory Relat. Fields 140(1–2), 195–205 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    H.J. Brascamp, E.H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions. Adv. Math. 20(2), 151–173 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    H.J. Brascamp, E.H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976)zbMATHCrossRefGoogle Scholar
  19. 19.
    H.J. Brascamp, E.H. Lieb, J.M. Luttinger, A general rearrangement inequality for multiple integrals. J. Funct. Anal. 17, 227–237 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    E.A. Carlen, Superadditivity of Fisher’s information and logarithmic Sobolev inequalities. J. Funct. Anal. 101(1), 194–211 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    M.H.M. Costa, T.M. Cover, On the similarity of the entropy power inequality and the Brunn-Minkowski inequality. IEEE Trans. Inf. Theory 30(6), 837–839 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    T.M. Cover, J.A. Thomas, Elements of Information Theory (Wiley, New York, 1991)zbMATHCrossRefGoogle Scholar
  23. 23.
    A. Dembo, T.M. Cover, J.A. Thomas, Information-theoretic inequalities. IEEE Trans. Inf. Theory 37(6), 1501–1518 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    A. Ehrhard, Symétrisation dans l’espace de Gauss. Math. Scand. 53(2), 281–301 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    N. Gozlan, C. Roberto, P.M. Samson, P. Tetali, Displacement convexity of entropy and related inequalities on graphs. Probab. Theory Relat. Fields 160(1–2), 47–94 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    N. Gozlan, C. Roberto, P.M. Samson, P. Tetali, Transport proofs of some discrete variants of the Prékopa-Leindler inequality (2019). arXiv preprint. arXiv:1905.04038Google Scholar
  27. 27.
    L. Gross, Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    A. Kechris, Classical Descriptive Set Theory, vol. 156 (Springer Science & Business Media, New York, 2012)zbMATHGoogle Scholar
  29. 29.
    J. Li, Rényi entropy power inequality and a reverse (2017). Preprint. arXiv:1704.02634Google Scholar
  30. 30.
    J. Li, J. Melbourne, Further investigations of the maximum entropy of the sum of two dependent random variables, in 2018 IEEE International Symposium on Information Theory (ISIT) (IEEE, Piscataway, 2018), pp. 1969–1972Google Scholar
  31. 31.
    J. Li, A. Marsiglietti, J. Melbourne, Entropic central limit theorem for rényi entropy, in 2018 IEEE International Symposium on Information Theory (ISIT) (IEEE, Piscataway, 2019)Google Scholar
  32. 32.
    J. Li, A. Marsiglietti, J. Melbourne, Further investigations of Rényi entropy power inequalities and an entropic characterization of s-concave densities, in Geometric Aspects of Functional Analysis – Israel Seminar (GAFA) 2017–2019. Lecture Notes in Mathematics, vol. 2256 (to appear)Google Scholar
  33. 33.
    J. Li, A. Marsiglietti, J. Melbourne, Rényi entropy power inequalities for s-concave densities, in 2019 IEEE International Symposium on Information Theory (ISIT) (IEEE, Piscataway, 2019)Google Scholar
  34. 34.
    E.H. Lieb, Proof of an entropy conjecture of Wehrl. Commun. Math. Phys. 62(1), 35–41 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    M. Madiman, J. Melbourne, P. Xu, Forward and reverse entropy power inequalities in convex geometry, in Convexity Concentration (Springer, New York, 2017), pp. 427–485zbMATHGoogle Scholar
  36. 36.
    M. Madiman, J. Melbourne, P. Xu, Rogozin’s convolution inequality for locally compact groups (2017). Preprint, arXiv:1705.00642Google Scholar
  37. 37.
    M. Madiman, L. Wang, J.O. Woo, Entropy inequalities for sums in prime cyclic groups (2017). arXiv preprint. arXiv:1710.00812Google Scholar
  38. 38.
    M. Madiman, L. Wang, J.O. Woo, Majorization and Rényi entropy inequalities via Sperner theory. Discrete Math. 342, 2911–2923 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    A. Marsiglietti, J. Melbourne, A Rényi entropy power inequality for log-concave vectors and parameters in [0, 1], in 2018 IEEE International Symposium on Information Theory (ISIT) (IEEE, Piscataway, 2018), pp. 1964–1968Google Scholar
  40. 40.
    A. Marsiglietti, J. Melbourne, On the entropy power inequality for the Rényi entropy of order [0, 1]. IEEE Trans. Inf. Theory 65, 1387–1396 (2019)zbMATHCrossRefGoogle Scholar
  41. 41.
    J. Martín, M. Milman, Isoperimetry and symmetrization for logarithmic Sobolev inequalities. J. Funct. Anal. 256(1), 149–178 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    J. Martín, M. Milman, Pointwise symmetrization inequalities for Sobolev functions and applications. Adv. Math. 225(1), 121–199 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    J. Melbourne, Rearrangements and information theoretic inequalities, in 2018 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton) (IEEE, Piscataway, 2018)Google Scholar
  44. 44.
    Y. Ollivier, C. Villani, A curved Brunn–Minkowski inequality on the discrete hypercube, or: what is the Ricci curvature of the discrete hypercube? SIAM J. Discrete Math. 26(3), 983–996 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    E. Ram, I. Sason, On rényi entropy power inequalities. IEEE Trans. Inf. Theory 62(12), 6800–6815 (2016)zbMATHCrossRefGoogle Scholar
  46. 46.
    A. Rényi, On measures of entropy and information, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1 (University of California Press, Berkeley, 1961), pp. 547–561Google Scholar
  47. 47.
    C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    A.J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inf. Control 2, 101–112 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    V.N. Sudakov, B.S. Tsirel’son, Extremal properties of half-spaces for spherically invariant measures. Zap. Nauch. Sem. LOMI 41, 14–24 (1974). Translated in J. Soviet Math. 9, 9–18 (1978)Google Scholar
  50. 50.
    L. Wang, M. Madiman, Beyond the entropy power inequality, via rearrangements. IEEE Trans. Inf. Theory 60(9), 5116–5137 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    R. Zamir, M. Feder, A generalization of the entropy power inequality with applications. IEEE Trans. Inf. Theory 39(5), 1723–1728 (1993)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Electrical and Computer EngineeringUniversity of MinnesotaMinneapolisUSA

Personalised recommendations