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

We give a general inequality for Bergman kernels of Bergman spaces defined by convex weights in ${\text{C}}^n$. We also discuss how this can be used in Nazarov’s proof of the Bourgain-Milman theorem, as a substitute for H"ormander’s estimates for the $\text{dbar}$-equation.