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Transport Proofs Of Some Discrete Variants Of The Pr{e}Kopa-leindler Inequality

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 Added by Paul-Marie Samson
 Publication date 2019
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and research's language is English




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We give a transport proof of a discrete version of the displacement convexity of entropy on integers (Z), and get, as a consequence, two discrete forms of the Pr{e}kopa-Leindler Inequality : the Four Functions Theorem of Ahlswede and Daykin on the discrete hypercube [1] and a recent result on Z due to Klartag and Lehec [16].



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112 - Yuchi Wu 2020
In this paper, we prove a Prekopa-Leindler type inequality of the $L_p$ Brunn-Minkowski inequality. It extends an inequality proved by Das Gupta [8] and Klartag [16], and thus recovers the Prekopa-Leindler inequality. In addition, we prove a functional $L_p$ Minkowski inequality.
In this paper, we present a simple proof of a recent result of the second author which establishes that functional inverse-Santal{o} inequalities follow from Entropy-Transport inequalities. Then, using transport arguments together with elementary correlation inequalities, we prove these sharp Entropy-Transport inequalities in dimension 1. We also revisit the proof of the functional inverse-Santal{o} inequalities in the n dimensional unconditional case using these ideas.
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