# Displacement convexity of entropy and related inequalities on graphs

@article{Gozlan2012DisplacementCO, title={Displacement convexity of entropy and related inequalities on graphs}, author={Nathael Gozlan and Cyril Roberto and Paul-Marie Samson and Prasad Tetali}, journal={Probability Theory and Related Fields}, year={2012}, volume={160}, pages={47-94} }

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 Prékopa-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…

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