  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 the recent work of Fradelizi, Madiman and Wang (2016). In
particular, our results imply that convex measures in high dimensions 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 R\'enyi
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 (1947).
