We establish sharp exponential deviation estimates of the information content as well as a sharp bound on the varentropy for the class of convex measures on Euclidean spaces. This generalizes a similar development for log-concave measures in recent work of Fradelizi, Madiman and Wang (2016). In particular, our results imply that convex measures in high dimension are concentrated in an annulus between two convex sets (as in the log-concave case) despite their possibly having much heavier tails. Various tools and consequences are developed, including a sharp comparison result for Rényi entropies, inequalities of Kahane-Khinchine-type for convex measures that extend those of Koldobsky, Pajor and Yaskin (2008) for log-concave measures, and an extension of Berwald's inequality. * Some of these results were announced at ISIT 2016 [24].