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For Polish spaces, we prove its equivalent entropic formulation using the Legendre-Fenchel duality theory. Capitalizing on the entropic formulation, we elaborate on a \u201cdoubling trick\u201d used by Lieb and Geng-Nair to prove the Gaussian optimality in this inequality for the case of Gaussian reference measures.<\/jats:p>","DOI":"10.3390\/e20060418","type":"journal-article","created":{"date-parts":[[2018,5,31]],"date-time":"2018-05-31T03:07:42Z","timestamp":1527736062000},"page":"418","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["A Forward-Reverse Brascamp-Lieb Inequality: Entropic Duality and Gaussian Optimality"],"prefix":"10.3390","volume":"20","author":[{"given":"Jingbo","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas A.","family":"Courtade","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720-1770, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Paul W.","family":"Cuff","sequence":"additional","affiliation":[{"name":"Renaissance Technologies, LLC 600 Route 25A East Setauket, New York, NY 11733, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sergio","family":"Verd\u00fa","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,5,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/0001-8708(76)90184-5","article-title":"Best constants in Young\u2019s inequality, its converse, and its generalization to more than three functions","volume":"20","author":"Brascamp","year":"1976","journal-title":"Adv. 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