Open Access
August 2018 Deviation of polynomials from their expectations and isoperimetry
Lavrentin M. Arutyunyan, Egor D. Kosov
Bernoulli 24(3): 2043-2063 (August 2018). DOI: 10.3150/16-BEJ919


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.


Download Citation

Lavrentin M. Arutyunyan. Egor D. Kosov. "Deviation of polynomials from their expectations and isoperimetry." Bernoulli 24 (3) 2043 - 2063, August 2018.


Received: 1 April 2016; Revised: 1 November 2016; Published: August 2018
First available in Project Euclid: 2 February 2018

zbMATH: 06839259
MathSciNet: MR3757522
Digital Object Identifier: 10.3150/16-BEJ919

Keywords: Carbery–Wright inequality , distribution of a polynomial , Gaussian measure , Isoperimetric inequality , logarithmically concave measure , Poincaré inequality

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 3 • August 2018
Back to Top