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Higher Lelong Numbers and Convex Geometry

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Abstract

We prove the reversed Alexandrov–Fenchel inequality for mixed Monge–Ampère masses of plurisubharmonic functions, which generalizes a result of Demailly and Pham. As applications to convex geometry, this gives a complex analytic proof of the reversed Alexandrov–Fenchel inequality for mixed covolumes, which generalizes recent results in convex geometry of Kaveh–Khovanskii, Khovanskii–Timorin, Milman–Rotem and Schneider on reversed (or complemented) Brunn–Minkowski and Alexandrov–Fenchel inequalities. Also for toric plurisubharmonic functions in the Cegrell class, we confirm Demailly’s conjecture on the convergence of higher Lelong numbers under the canonical approximation.

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Acknowledgements

We would like to thank Jean-Pierre Demailly and Sébastien Boucksom for interesting discussions regarding Conjecture 1.1, and Kiumars Kaveh, Askold Khovanskii and Vladlen Timorin for answering our questions on their papers. We thank the anonymous referees for helpful comments. This work began when A.R. visited Seoul National University and D.K. visited University of Stavanger and École Polytechnique. We thank these institutions for hospitality and also SRC-GAIA (Center for Geometry and its Applications, based at POSTECH, Korea) for its financial support for these visits through the National Research Foundation of Korea grant No.2011-0030795.

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  1. Department of Mathematical Sciences & Research Institute of Mathematics, Seoul National University, Seoul, 08826, Korea

    Dano Kim

  2. Tek/Nat, University of Stavanger, 4036, Stavanger, Norway

    Alexander Rashkovskii

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  1. Dano Kim
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Correspondence to Dano Kim.

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Kim, D., Rashkovskii, A. Higher Lelong Numbers and Convex Geometry. J Geom Anal 31, 2525–2539 (2021). https://doi.org/10.1007/s12220-020-00362-w

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