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arXiv · 1207.5116

Displacement convexity of entropy and related inequalities on graphs

Nathaël Gozlan·Cyril Roberto·Paul-Marie Samson·Prasad Tetali
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Abstract

We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prekopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal -- by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.

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Nathaël Gozlan, Cyril Roberto, Paul-Marie Samson, Prasad Tetali. 2012-07-21. Displacement convexity of entropy and related inequalities on graphs. https://arxiv.org/abs/1207.5116

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