The Distribution of Values of Analytic Functions on Convex Bodies

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.