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We provide a mild sufficient condition for a probability measure on the realline to satisfy a modified log-Sobolev inequality for convex functions,interpolating between the classical log-Sobolev inequality and a Bobkov-Ledouxtype inequality. As a consequence we obtain dimension-free two-levelconcentration results for convex function of independent random variables withsufficiently regular tail decay. We also provide a link between modifiedlog-Sobolev inequalities for convex functions and weak transport-entropyinequalities, complementing recent work by Gozlan, Roberto, Samson, and Tetali.

Modified log-Sobolev inequalities for convex functions on the real line. Sufficient conditions