## Bernoulli

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

### Deviation of polynomials from their expectations and isoperimetry

Lavrentin M. Arutyunyan and Egor D. Kosov

#### 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**

Received: April 2016

Revised: November 2016

First available in Project Euclid: 2 February 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.3150/16-BEJ919

**Mathematical Reviews number (MathSciNet)**

MR3757522

**Zentralblatt MATH identifier**

06839259

**Keywords**

Carbery–Wright inequality distribution of a polynomial Gaussian measure isoperimetric inequality logarithmically concave measure Poincaré inequality

#### 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