This short note deals with elementary semigroup arguments together with Harris's negative association inequality in order to obtain improved deviation inequalities (for the left tail) for log-concave measures under some monotony and convexity assumption.
On the convex Poincaré inequality and weak transportation inequalities
Transport Inequalities. A Survey
Logarithmic Sobolev Inequalities and Concentration of Measure for Convex Functions and Polynomial Chaoses
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