Abstract
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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References
[G]L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97, 1061–1083 (1975).
[L]M. Ledoux, Isoperimétrie et inegalités de Sobolev logarithmiques gaussiennes, C. R. Acad. Sci. Paris 306, 79–82 (1988).
[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.
[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).
[R]L. Russo, An approximate zero-law, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandete Gebeite 61, 129–139 (1982).
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Work partially supported by the US-Israel Binational Science Foundation.
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Talagrand, M. Isoperimetry, logarithmic sobolev inequalities on the discrete cube, and margulis' graph connectivity theorem. Geometric and Functional Analysis 3, 295–314 (1993). https://doi.org/10.1007/BF01895691
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DOI: https://doi.org/10.1007/BF01895691