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

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

References in corpus (12)

• A direct proof of the functional Santalo inequality
• 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
• Combinatorial Entropy Power Inequalities: A Preliminary Study of the Stam region
• Near Equality in the Brunn-Minkowski Inequality
• Near equality in the two-dimensional Brunn-Minkowski inequality
• Wasserstein Stability of the Entropy Power Inequality for Log-Concave Densities
• Rényi Divergence and $L_p$-affine surface area for convex bodies
• The convexification effect of Minkowski summation
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Cited by in corpus (22)

• 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
• Entropy Inequalities for Sums in Prime Cyclic Groups
• Rényi Entropy Power Inequalities via Normal Transport and Rotation
• Majorization and Rényi Entropy Inequalities via Sperner Theory
• 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
• Links between the Logarithmic Sobolev Inequality and the convolution inequalities for Entropy and Fisher Information
• Bernoulli sums and Rényi entropy inequalities
• Error Bounds on a Mixed Entropy Inequality
• Conditional Rényi entropy and the relationships between Rényi capacities
• Volumes of subset Minkowski sums and the Lyusternik region
• 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
• Rényi Entropy Power and Normal Transport
• Entropies of sums of independent gamma random variables
• Weighted $p$-Rényi Entropy Power Inequality: Information Theory to Quantum Shannon Theory
• The norm of the Fourier transform on compact or discrete abelian groups