From log Sobolev to Talagrand: A quick proof
Nicola Gigli Michel Ledoux
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 1927-1935 doi: 10.3934/dcds.2013.33.1927
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.
keywords: log-Sobolev inequality Talagrand inequality. Metric measure spaces

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