Concentration of information content for convex measures

Abstract

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 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 (1947).

Added some references

References in corpus (3)

• Entropy bounds on abelian groups and the Ruzsa divergence
• Dimensional behaviour of entropy and information
• Do Minkowski averages get progressively more convex?

Cited by in corpus (5)

• The Differential Entropy of Mixtures: New Bounds and Applications
• Sharp moment-entropy inequalities and capacity bounds for log-concave distributions
• Affine Isoperimetric Inequalities for Higher-Order Projection and Centroid Bodies
• Conditional Rényi entropy and the relationships between Rényi capacities
• Entropic exercises around the Kneser-Poulsen conjecture