Concentration inequalities for $s$-concave measures of dilations of Borel sets and applications

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Fradelizi, Matthieu
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We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel sets in $\mathbb{R}^n$ by a $s$-concave probability. Our result gives a common generalization of an inequality of Nazarov, Sodin and Volberg and a concentration inequality of Gu\'edon. Applying our inequality to the level sets of functions satisfying a Remez type inequality, we deduce, as it is classical, that these functions enjoy dimension free distribution inequalities and Kahane-Khintchine type inequalities with positive and negative exponent, with respect to an arbitrary $s$-concave probability.
Comment: 22 pages, submitted
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Mathematics - Probability, Mathematics - Functional Analysis, 46B07, 46B09, 60B11, 52A20, 26D05
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