Forward and Reverse Entropy Power Inequalities in Convex Geometry

Abstract

The entropy power inequality, which plays a fundamental role in information theory and probability, may be seen as an analogue of the Brunn-Minkowski inequality. Motivated by this connection to Convex Geometry, we survey various recent developments on forward and reverse entropy power inequalities not just for the Shannon-Boltzmann entropy but also more generally for Rényi entropy. In the process, we discuss connections between the so-called functional (or integral) and probabilistic (or entropic) analogues of some classical inequalities in geometric functional analysis

arXiv comment54 pages. Changes in v2: improved organization, cleaned up exposition, and numerous references added

References in corpus (38)

• Generalized Entropy Power Inequalities and Monotonicity Properties of Information
• Information Theoretic Proofs of Entropy Power Inequalities
• Fisher Information inequalities and the Central Limit Theorem
• Sumset and inverse sumset theorems for Shannon entropy
• The entropy per coordinate of a random vector is highly constrained under convexity conditions
• Log-concavity and the maximum entropy property of the Poisson distribution
• Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures
• Beyond the entropy power inequality, via rearrangements
• Concentration of the information in data with log-concave distributions
• Some results concerning maximum Renyi entropy distributions
• The concavity of Rènyi entropy power
• A direct proof of the functional Santalo inequality
• Optimal Concentration of Information Content For Log-Concave Densities
• Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures
• Compound Poisson Approximation via Information Functionals
• Entropy bounds on abelian groups and the Ruzsa divergence
• On the problem of reversibility of the entropy power inequality
• Entropy and set cardinality inequalities for partition-determined functions
• Dimensional behaviour of entropy and information
• Partitions and functional Santalo inequalities
• On an Inequality of Karlin and Rinott Concerning Weighted Sums of i.i.d. Random Variables
• Rogozin's convolution inequality for locally compact groups
• Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality
• A conditional Entropy Power Inequality for dependent variables
• Near Equality in the Brunn-Minkowski Inequality
• Monotonic Convergence in an Information-Theoretic Law of Small Numbers
• Combinatorial Entropy Power Inequalities: A Preliminary Study of the Stam region
• The Blaschke-Santalo Inequality
• Near equality in the two-dimensional Brunn-Minkowski inequality
• Near-extremizers of Young's Inequality for R^d
• Do Minkowski averages get progressively more convex?
• Wasserstein Stability of the Entropy Power Inequality for Log-Concave Densities
• Arak Inequalities for Concentration Functions and the Littlewood-Offord Problem
• Rényi Divergence and $L_p$-affine surface area for convex bodies
• A combinatorial approach to small ball inequalities for sums and differences
• The convexification effect of Minkowski summation
• α-concave functions and a functional extension of mixed volumes
• An information-theoretic Central Limit Theorem for finitely susceptible FKG systems

Cited by in corpus (27)

• Entropy bounds on abelian groups and the Ruzsa divergence
• A lower bound on the differential entropy of log-concave random vectors with applications
• Rogozin's convolution inequality for locally compact groups
• Combinatorial Entropy Power Inequalities: A Preliminary Study of the Stam region
• Rényi Entropy Power Inequalities via Normal Transport and Rotation
• Entropy Inequalities for Sums in Prime Cyclic Groups
• Majorization and Rényi Entropy Inequalities via Sperner Theory
• Bounds on Variance for Unimodal Distributions
• The Differential Entropy of Mixtures: New Bounds and Applications
• Sharp moment-entropy inequalities and capacity bounds for log-concave distributions
• On the sine polarity and the $L_p$-sine Blaschke-Santaló inequality
• Two remarks on generalized entropy power inequalities
• Bernoulli sums and Rényi entropy inequalities
• Links between the Logarithmic Sobolev Inequality and the convolution inequalities for Entropy and Fisher Information
• Volumes of subset Minkowski sums and the Lyusternik region
• Conditional Rényi entropy and the relationships between Rényi capacities
• Error Bounds on a Mixed Entropy Inequality
• Entropies of weighted sums in cyclic groups and an application to polar codes
• A combinatorial approach to small ball inequalities for sums and differences
• Information Inequalities via Submodularity and a Problem in Extremal Graph Theory
• Reversals of Rényi Entropy Inequalities under Log-Concavity
• Entropic exercises around the Kneser-Poulsen conjecture
• Entropy and thinning of discrete random variables
• Entropies of sums of independent gamma random variables
• Rényi Entropy Power and Normal Transport
• The norm of the Fourier transform on compact or discrete abelian groups
• Weighted $p$-Rényi Entropy Power Inequality: Information Theory to Quantum Shannon Theory