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

Part of the Lecture Notes in Mathematics book series (LNM,volume 2116)

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.

Keywords

  • Bergman Kernel
  • Pluricomplex Green Function
  • Suita Conjecture
  • Kobayashi Indicatrix
  • Sublevel Sets

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Zbigniew Błocki .

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Błocki, Z. (2014). A Lower Bound for the Bergman Kernel and the Bourgain-Milman Inequality. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_4

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