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On the convex Poincaré inequality and weak transportation inequalities. (English) Zbl 1442.60030
Summary: We prove that for a probability measure on \(\mathbb{R}^{n}\), the Poincaré inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This generalizes recent results by Gozlan, Roberto, Samson, Shu, Tetali and Feldheim, Marsiglietti, Nayar, Wang, concerning probability measures on the real line.
The proof relies on modified logarithmic Sobolev inequalities of Bobkov-Ledoux type for convex and concave functions, which are of independent interest.
We also present refined concentration inequalities for general (not necessarily Lipschitz) convex functions, complementing recent results by S. G. Bobkov et al. [Lect. Notes Math. 2169, 25–53 (2017; Zbl 1366.60052)].

MSC:
60E15 Inequalities; stochastic orderings
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
60D05 Geometric probability and stochastic geometry
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