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

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.