Deviation of polynomials from their expectations and isoperimetry

TL;DR

This paper investigates the deviation of polynomials from their expectations, providing new lower bounds for Gaussian and log-concave measures, and explores isoperimetric and Poincaré inequalities for polynomial images of convex sets.

Contribution

It introduces novel lower deviation estimates for polynomials under Gaussian and log-concave measures and studies isoperimetric and Poincaré inequalities for polynomial images of convex sets.

Findings

Established lower bounds for polynomial deviations under Gaussian measures.

Derived deviation estimates for degree-two polynomials in log-concave measures.

Analyzed isoperimetric and Poincaré inequalities for polynomial images of convex sets.

Abstract

In the first part we study 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 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 deals with isoperimetric inequality and the Poincar\'e 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.