Abstract
We introduce a concentration property for probability measures onR n, which we call Property (τ); we show that this property has an interesting stability under products and contractions (Lemmas 1, 2, 3). Using property (τ), we give a short proof for a recent deviation inequality due to Talagrand. In a third section, we also recover known concentration results for Gaussian measures using our approach.
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References
[B]C. Borell, The Brunn-Minkowski inequality in Gauss space, Inventiones Math. 30 (1975), 205–216.
[C]L. Chen, An inequality for the multivariate normal distribution, J. Multivariate Anal. 12 (1982) 306–315.
[E]A. Ehrhard, Symétrisation dans l'espace de Gauss, Math. Scand. 53 (1983) 281–301.
[FLM]T. Figiel, J. Lindenstrauss, V. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977) 53–94.
[JS]W. Johnson, G. Schechtman, Remarks on Talagrand's deviation inequality for Rademacher functions, Texas Functional Analysis Seminar 1988–89.
[L]L. Leindler, On a certain converse of Hölder's inequality, Acta Sci. Math. 33 (1972) 217–223.
[M]V. Milman, A new proof of the theorem of A. Dvoretzky on sections of convex bodies, Func. Anal. Appl. 5 (1971) 28–37.
[MS]V. Milman, G. Schechtman, Asymptotic theory of finite dimensional normed spaces, Springer Lecture Notes in Math. 1200 (1986).
[Pi1]G. Pisier, Probabilistic methods in the geometry of Banach spaces, CIME Varenna 1985, Springer Lecture Notes in Math. 1206, 167–241.
[Pi2]G. Pisier, Volume of convex bodies and Banach spaces geometry, Cambridge University Press.
[Pr]A. Prekopa, On logarithmically concave measures and functions, Acta Sci. Math. 34 (1973) 335–343.
[ST]V. Sudakov, B. Tsirelson, Extremal properties of half spaces for spherically invariant measures, Zap. Nauch. Sem. LOMI 41, (1974) 14–24 translated in J. Soviet Math. 9 (1978) 9–18.
[Ta1]M. Talagrand, Seminar lecture in Paris, Spring 1990.
[Ta2]M. Talagrand, A new isoperimetric inequality and the concentration of measure phenomenon, to appear in GAFA90, Springer LNM (1991), 94–124.
[Ta3]M. Talagrand, An isoperimetric theorem on the cube and the Khintchine-Kahane inequalities, Proc. Amer. Math. Soc. 104 (1988) 905–909.
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Maurey, B. Some deviation inequalities. Geometric and Functional Analysis 1, 188–197 (1991). https://doi.org/10.1007/BF01896377
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DOI: https://doi.org/10.1007/BF01896377