• Bernoulli
  • Volume 25, Number 1 (2019), 341-374.

On the convex Poincaré inequality and weak transportation inequalities

Radosław Adamczak and Michał Strzelecki

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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.

Article information

Bernoulli, Volume 25, Number 1 (2019), 341-374.

Received: March 2017
Revised: July 2017
First available in Project Euclid: 12 December 2018

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concentration of measure convex functions Poincaré inequality weak transport-entropy inequalities


Adamczak, Radosław; Strzelecki, Michał. On the convex Poincaré inequality and weak transportation inequalities. Bernoulli 25 (2019), no. 1, 341--374. doi:10.3150/17-BEJ989.

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