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Johnson","submitted_at":"1990-02-16T00:00:00Z","abstract_excerpt":"Recently Talagrand [T] estimated the deviation of a function on $\\{0,1\\}^n$ from its median in terms of the Lipschitz constant of a convex extension of $f$ to $\\ell ^n_2$; namely, he proved that\n  $$P(|f-M_f| > c) \\le 4 e^{-t^2/4\\sigma ^2}$$ where $\\sigma$ is  the Lipschitz constant of the extension of $f$ and $P$ is the natural probability on $\\{0,1\\}^n$.\n  Here we extend this inequality to more general product probability spaces; in particular, we prove the same inequality for $\\{0,1\\}^n$ with the product measure $((1-\\eta)\\delta _0 + \\eta \\delta _1)^n$. 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