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We give a general inequality for Bergman kernels of Bergman spaces defined byconvex weights in $\C^n$. We also discuss how this can be used in Nazarov'sproof of the Bourgain-Milman theorem, as a substitute for H\"ormander'sestimates for the $\dbar$-equation.

Bergman kernels for Paley-Wiener spaces and Nazarov's proof of the Bourgain-Milman theorem