Geometric & Functional Analysis GAFA

, Volume 3, Issue 3, pp 295–314

Isoperimetry, logarithmic sobolev inequalities on the discrete cube, and margulis' graph connectivity theorem

  • M. Talagrand


We prove that the suitably defined surface area of a subsetA of the cube {0,1}n is bounded below by a certain explicit function of the size ofA. We establish a family of logarithmic Sobolev inequalities on the cube related to this isoperimetric result. We also give a quantitative version of Margulis' graph connectivity theorem.


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  1. [G]L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97, 1061–1083 (1975).Google Scholar
  2. [L]M. Ledoux, Isoperimétrie et inegalités de Sobolev logarithmiques gaussiennes, C. R. Acad. Sci. Paris 306, 79–82 (1988).Google Scholar
  3. [M]G.A. Margulis, Probabilistic characteristics of graphs with large connectivity, Problemy Peredachi Informatsii 10:2 (1974), 101–108; English translation: Problems Info. Transmission 10 (1977), 174–179, Plenum Press, New York.Google Scholar
  4. [P]G. Pisier, Probabilistic methods in the geometry of Banach spaces, in “Probability and Analysis, Varenna (Italy) 1985”, Lecture Notes in Math. 1206, 167–241, Springer Verlag (1986).Google Scholar
  5. [R]L. Russo, An approximate zero-law, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandete Gebeite 61, 129–139 (1982).Google Scholar

Copyright information

© Birkhäuser Verlag 1993

Authors and Affiliations

  • M. Talagrand
    • 1
    • 2
  1. 1.Equipe d'Analyse-Tour 48 U.A. au C.N.R.S. no. 745Université Paris VIParis Cedex 05France
  2. 2.Department of MathematicsOhio State UniversityColumbusUSA

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