Bernoulli

  • 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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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
Received: March 2017
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

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

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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