The Distribution of Values of Analytic Functions on Convex Bodies

TL;DR

This paper establishes new dimensionless inequalities for analytic functions on convex bodies using Chebyshev degree, leading to reverse Holder inequalities with universal constants, extending recent polynomial results to broader analytic functions.

Contribution

It introduces novel inequalities for analytic functions based on Chebyshev degree and generalizes reverse Holder inequalities with absolute constants.

Findings

Derived new inequalities for analytic functions on convex bodies.

Extended reverse Holder inequalities to analytic functions with universal constants.

Connected inequalities for polynomials to broader classes of analytic functions.

Abstract

Proceeding the study of local properties of analytic functions started in [Br] we prove new dimensionless inequalities for such functions in terms of their Chebyshev degree. As a consequence, we obtain the reverse Holder inequalities for analytic functions with absolute (i.e., independent of dimension) constants. For polynomials such inequalities were recently proved by Bobkov who sharpened and generalized the previous Bourgain result and by Sodin and Volberg.