A characterization of a class of convex log-Sobolev inequalities on the real line. (English. French summary) Zbl 1403.60022

Summary: We give a sufficient and necessary condition for a probability measure \(\mu\) on the real line to satisfy the logarithmic Sobolev inequality for convex functions. The condition is expressed in terms of the unique left-continuous and non-decreasing map transporting the symmetric exponential measure onto \(\mu\). The main tool in the proof is the theory of weak transport costs.
As a consequence, we obtain dimension-free concentration bounds for the lower and upper tails of convex functions of independent random variables which satisfy the convex log-Sobolev inequality.


60E15 Inequalities; stochastic orderings
26A51 Convexity of real functions in one variable, generalizations
26D10 Inequalities involving derivatives and differential and integral operators
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