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On Nazarov’s Complex Analytic Approach to the Mahler Conjecture and the Bourgain-Milman Inequality

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Complex Analysis and Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 144))

Abstract

We survey the several complex variables approach to the Mahler conjecture from convex analysis due to Nazarov. We also show, although only numerically, that his proof of the Bourgain-Milman inequality using estimates for the Bergman kernel for tube domains cannot be improved to obtain the Mahler conjecture which would be the optimal version of this inequality.

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References

  1. Berndtsson, B.: Weighted estimates for the \(\overline{\partial }\)-equation, Complex Analysis and Geometry, Columbus, Ohio, 1999, Ohio State Univ. Math. Res. Inst. Publ. 9, pp. 43–57, Walter de Gruyter (2001)

    Google Scholar 

  2. Berndtsson, B.: Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains. Ann. Inst. Fourier 56, 1633–1662 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blaschke, W.: Über affine Geometrie VII: Neue Extremeigenschaften von Ellipse und Ellipsoid, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl. 69, 306–318 (1917)

    Google Scholar 

  4. Blaschke, W.: Affine Geometrie IX: Verschiedene Bemerkungen und Aufgaben, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl. 69, 412–420 (1917)

    Google Scholar 

  5. Błocki, Z.: Suita conjecture and the Ohsawa-Takegoshi extension theorem. Invent. Math. 193, 149–158 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Błocki, Z.: A lower bound for the Bergman kernel and the Bourgain-Milman inequality, Geometric Aspects of Functional Analysis, Israel Seminar (GAFA) 2011–2013. In: Klartag, B., Milman, E. (eds.) Lecture Notes in Mathematics, vol. 2116, pp. 53–63. Springer (2014)

    Google Scholar 

  7. Błocki, Z.: Cauchy-Riemann meet Monge-Ampère. Bull. Math. Sci. 4, 433–480 (2014)

    Article  MathSciNet  Google Scholar 

  8. Błocki, Z., Zwonek, W.: Estimates for the Bergman kernel and the multidimensional Suita conjecture. New York J. Math. 21, 151–161 (2015)

    Google Scholar 

  9. Błocki, Z., Zwonek, W.: On the Suita conjecture for some convex ellipsoids in \(\mathbb{C}^{2}\). arXiv:1409.5023, Experimental Math. (to appear)

  10. Bourgain, J., Milman, V.: New volume ratio properties for convex symmetric bodies in \(\mathbb{R}^n\). Invent. Math. 88, 319–340 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. Donnelly, H., Fefferman, C.: \(L^2\)-cohomology and index theorem for the Bergman metric. Ann. of Math. 118, 593–618 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hansen, A.B., Lima, Å.: The structure of finite-dimensional Banach spaces with the 3.2. intersection property. Acta Math. 146, 1–23 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hörmander, L.: \(L^{2}\) estimates and existence theorems for the \(\bar{\partial }\) operator. Acta Math. 113, 89–152 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hsin, C.-I.: The Bergman kernel on tube domains. Rev. Un. Mat. Argentina 46, 23–29 (2005)

    MathSciNet  MATH  Google Scholar 

  15. Kuperberg, G.: From the Mahler conjecture to Gauss linking integrals. Geom. Funct. Anal. 18, 870–892 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lempert, L.: La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. Fr. 109, 427–474 (1981)

    MathSciNet  MATH  Google Scholar 

  17. Lempert, L.: Private Communication, October (2013)

    Google Scholar 

  18. Mahler, K.: Ein Minimalproblem für konvexe Polygone. Mathematika B (Zutphen) 7, 118–127 (1938)

    Google Scholar 

  19. Nazarov, F.: The Hörmander proof of the Bourgain-Milman theorem. In: Klartag, B., Mendelson, S., Milman, V.D. (eds.) Geometric Aspects of Functional Analysis, Israel Seminar 2006–2010. Lecture Notes in Mathematics, vol. 2050, pp. 335–343. Springer (2012)

    Google Scholar 

  20. Rothaus, O.S.: Some properties of Laplace transforms of measures. Trans. Amer. Math. Soc. 131, 163–169 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ryabogin, D., Zvavitch, A.: Analytic methods in convex geometry. Lectures given at the Polish Academy of Sciences, November 2011, IMPAN Lecture Notes (to appear). http://www.impan.pl/Dokt/EN/SpLect/RZ2011.pdf

  22. Saint-Raymond, J.: Sur le volume des corps convexes symétriques. In: Initiation Seminar on Analysis: G. Choquet, M. Rogalski, J. Saint-Raymond, 20th Year: 1980/1981, Exp. No. 11, pp. 25, Publ. Math. Univ. Pierre et Marie Curie, 46, Univ. Paris VI, Paris (1981)

    Google Scholar 

  23. Santaló, L.A.: An affine invariant for convex bodies of n-dimensional space. Portugaliae Math. 8, 155–161 (1949). (in Spanish)

    MathSciNet  MATH  Google Scholar 

  24. Suita, N.: Capacities and kernels on Riemann surfaces. Arch. Ration. Mech. Anal. 46, 212–217 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zajac, S.: Complex geodesics in convex tube domains. arXiv:1303.0014, Ann. Scuola Norm. Sup. Pisa (to appear)

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Acknowledgments

Partially supported by the Ideas Plus grant 0001/ID3/2014/63 of the Polish Ministry of Science and Higher Education.

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Authors and Affiliations

  1. Uniwersytet Jagielloński, Instytut Matematyki, Łojasiewicza 6, 30-348, Kraków, Poland

    Zbigniew Błocki

Authors
  1. Zbigniew Błocki
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Correspondence to Zbigniew Błocki .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Università di Roma Tor Vergata, Rome, Italy

    Filippo Bracci

  2. Dept. of Mathematics Education, Kyungnam University, Changwon, Korea (Republic of)

    Jisoo Byun

  3. Université Joseph Fourier Grenoble, Institut Fourier, Grenoble, France

    Hervé Gaussier

  4. Graduate School of Mathematical Sciences, The University of Tokyo, Meguro-ku, Tokyo, Japan

    Kengo Hirachi

  5. Department of Mathematics, POSTECH, Pohang, Korea (Republic of)

    Kang-Tae Kim

  6. Department of Mathematics, Bergische Universität Wuppertal, Wuppertal, Nordrhein-Westfalen, Germany

    Nikolay Shcherbina

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Błocki, Z. (2015). On Nazarov’s Complex Analytic Approach to the Mahler Conjecture and the Bourgain-Milman Inequality. In: Bracci, F., Byun, J., Gaussier, H., Hirachi, K., Kim, KT., Shcherbina, N. (eds) Complex Analysis and Geometry. Springer Proceedings in Mathematics & Statistics, vol 144. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55744-9_6

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