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 dilationsof Borel sets in $\mathbb{R}^n$ by a $s$-concave probability. Our result givesa common generalization of an inequality of Nazarov, Sodin and Volberg and aconcentration inequality of Gu\'edon. Applying our inequality to the level setsof functions satisfying a Remez type inequality, we deduce, as it is classical,that these functions enjoy dimension free distribution inequalities andKahane-Khintchine type inequalities with positive and negative exponent, withrespect to an arbitrary $s$-concave probability.