# American Institute of Mathematical Sciences

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May  2013, 33(5): 1927-1935. doi: 10.3934/dcds.2013.33.1927

## From log Sobolev to Talagrand: A quick proof

 1 Laboratoire J. A. Dieudonné, Université de Nice, Parc Valrose, 06108 Nice, France 2 Institut de Mathématiques de Toulouse, Université de Toulouse, 31062 Toulouse, France

Received  December 2011 Revised  February 2012 Published  December 2012

We provide yet another proof of the Otto-Villani theorem from the log Sobolev inequality to the Talagrand transportation cost inequality valid in arbitrary metric measure spaces. The argument relies on the recent development [2] identifying gradient flows in Hilbert space and in Wassertein space, emphasizing one key step as precisely the root of the Otto-Villani theorem. The approach does not require the doubling property or the validity of the local Poincaré inequality.
Citation: Nicola Gigli, Michel Ledoux. From log Sobolev to Talagrand: A quick proof. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1927-1935. doi: 10.3934/dcds.2013.33.1927
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##### References:
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