Measure Concentration and Geometric Functional Analysis

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120sadvanced (9/10)state theorem

A probability measure μ on R^{d} satisfies a log-Sobolev inequality with constant c if Ent_{μ} (f^{2}) \leq 2 c \int ∣\nabla f ∣^{2} d μ for all smooth f, where Ent_{μ} (g) = \int g lo g g d μ - \int g d μ \cdot lo g \int g d μ . The Herbst argument shows this implies sub-Gaussian concentration. What is the Herbst argument's output, and how does the log-Sobolev constant c enter?