Integral Remez inequalities for polynomials on convex bodies

Abstract

We denote as an integral Remez inequality an inequality of the form $$\Vert f{\Vert }_{L^1(\mu )}\le C(\Omega ,\mu (A),X)\Vert f{\Vert }_{L^1({\mu }_A)},$$ 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.

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