Integral Remez inequalities for polynomials on convex bodies

TL;DR

This paper establishes integral Remez inequalities for polynomials over convex bodies, providing bounds that are independent of the dimension, contrasting with classical supremum-based inequalities.

Contribution

The paper introduces dimension-independent integral Remez inequalities for polynomials on convex bodies under uniform measures.

Findings

Dimension-independent bounds for polynomial integrals over convex bodies.

Contrast with classical L-infinity Remez inequalities.

Applicable to polynomials of arbitrary degree.

Abstract

We denote as an integral Remez inequality an inequality of the form $$ \|f\|_{L^{1}(\mu)} \le C(\Omega,\mu(A), X) \|f\|_{L^{1}(\mu_{A})}, $$ where $\mu_A$ is the normalised restriction of a measure $\mu$ to a set $A$. Let $\mu$ be the uniform distribution over a convex body A and $f$ be a polynomial of degree $d$. One can choose $C$ independent of the dimension of the set A, in contrast with a classical $L^{\infty}$ Remez inequality.