Open Access
February 2019 On the convex Poincaré inequality and weak transportation inequalities
Radosław Adamczak, Michał Strzelecki
Bernoulli 25(1): 341-374 (February 2019). DOI: 10.3150/17-BEJ989


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 Bobkov, Nayar, and Tetali.


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Radosław Adamczak. Michał Strzelecki. "On the convex Poincaré inequality and weak transportation inequalities." Bernoulli 25 (1) 341 - 374, February 2019.


Received: 1 March 2017; Revised: 1 July 2017; Published: February 2019
First available in Project Euclid: 12 December 2018

zbMATH: 07007210
MathSciNet: MR3892322
Digital Object Identifier: 10.3150/17-BEJ989

Keywords: concentration of measure , Convex functions , Poincaré inequality , Weak transport-entropy inequalities

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 1 • February 2019
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