A Lower Bound for the Bergman Kernel and the Bourgain-Milman Inequality

Abstract

For pseudoconvex domains in $\mathbb{C}^{n}$ we prove a sharp lower bound for the Bergman kernel in terms of volume of sublevel sets of the pluricomplex Green function. For n = 1 it gives in particular another proof of the Suita conjecture. If $\Omega$ is convex then by Lempert’s theory the estimate takes the form $K_{\Omega }(z) \geq 1/\lambda _{2n}(I_{\Omega }(z))$, where $I_{\Omega }(z)$ is the Kobayashi indicatrix at z. One can use this to simplify Nazarov’s proof of the Bourgain-Milman inequality from convex analysis. Possible further applications of Lempert’s theory in this area are also discussed.