## Bernoulli

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

### On the convex Poincaré inequality and weak transportation inequalities

#### Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1544605249

Digital Object Identifier
doi:10.3150/17-BEJ989

Mathematical Reviews number (MathSciNet)
MR3892322

Zentralblatt MATH identifier
07007210

#### Citation

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. https://projecteuclid.org/euclid.bj/1544605249

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