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 Renyi 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.

Forward and Reverse Entropy Power Inequalities in Convex Geometry

Over the past few years, a family of interesting new inequalities for the entropies of sums and differences of random variables has been developed by Ruzsa, Tao, and others, motivated by analogous results in additive combinatorics. This paper extends these earlier results to the case of random variables taking values in $ {\mathbb R}^{n}$ or, more generally, in arbitrary locally compact and Polish abelian groups. We isolate and study a key quantity, the Ruzsa divergence between two probability distributions, and we show that its properties can be used to extend the earlier inequalities to the present general setting. The new results established include several variations on the theme that the entropies of the sum and the difference of two independent random variables severely constrain each other. Although the setting is quite general, the results are already of interest (and new) for random vectors in $ {\mathbb R}^{n}$ . In that special case, we discuss quantitative bounds for the stability of the equality conditions in the entropy power inequality, a reverse entropy power inequality for log-concave random vectors, an information-theoretic analog of the Rogers–Shephard inequality for convex bodies, and consequences of some of our results to determinant inequalities for sums of positive-definite matrices. Moreover, by considering various multiplicative subgroups of the complex plane, one obtains new inequalities for the differential entropies of products and ratios of nonzero, complex-valued random variables.

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Entropy Bounds on Abelian Groups and the Ruzsa Divergence

We provide functional analogues of the classical geometric inequality of Rogers and Shephard on products of volumes of sections and projections. As a consequence we recover (and obtain some new) functional versions of Rogers–Shephard type inequalities as well as some generalizations of the geometric Rogers–Shephard inequality in the case where the subspaces intersect. These generalizations can be regarded as sharp local reverse Loomis–Whitney inequalities. We also obtain a sharp local Loomis–Whitney inequality.

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Rogers–Shephard and local Loomis–Whitney type inequalities

LYZ ellipsoid and Petty projection body for log-concave functions

We provide functional analogues of the classical geometric inequality of Rogers and Shephard on products of volumes of sections and projections. As a consequence we recover (and obtain some new) functional versions of Rogers-Shephard type inequalities as well as some generalizations of the geometric Rogers-Shephard inequality in the case where the subspaces intersect. These generalizations can be regarded as sharp local reverse Loomis-Whitney inequalities. We also obtain a sharp local Loomis-Whitney inequality.

Rogers-Shephard and local Loomis-Whitney type inequalities

https://zaguan.unizar.es/record/57928/files/texto_completo.pdf

B. González Merino

Papers

Rogers–Shephard inequality for log-concave functions