2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/163942We 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 Guédon. 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.22 pages, submittedProbabilityFunctional Analysis46B07; 46B09; 60B11; 52A20; 26D05text