  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.
