## Bernoulli

• Bernoulli
• Volume 24, Number 3 (2018), 2043-2063.

### Deviation of polynomials from their expectations and isoperimetry

#### Abstract

The article is divided into two parts. In the first part, we study the deviation of a polynomial from its mathematical expectation. This deviation can be estimated from above by Carbery–Wright inequality, so we investigate estimates of the deviation from below. We obtain such type estimates in two different cases: for Gaussian measures and a polynomial of an arbitrary degree and for an arbitrary log-concave measure but only for polynomials of the second degree. In the second part, we deal with the isoperimetric inequality and the Poincaré inequality for probability measures on the real line that are images of the uniform distributions on convex compact sets in $\mathbb{R}^{n}$ under polynomial mappings.

#### Article information

Source
Bernoulli, Volume 24, Number 3 (2018), 2043-2063.

Dates
Revised: November 2016
First available in Project Euclid: 2 February 2018

https://projecteuclid.org/euclid.bj/1517540467

Digital Object Identifier
doi:10.3150/16-BEJ919

Mathematical Reviews number (MathSciNet)
MR3757522

Zentralblatt MATH identifier
06839259

#### Citation

Arutyunyan, Lavrentin M.; Kosov, Egor D. Deviation of polynomials from their expectations and isoperimetry. Bernoulli 24 (2018), no. 3, 2043--2063. doi:10.3150/16-BEJ919. https://projecteuclid.org/euclid.bj/1517540467

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