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Higher Lelong numbers and convex geometry

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arxiv 1803.07948 v5 pith:KWP4LIPS submitted 2018-03-21 math.CV math.AGmath.MG

classification math.CVmath.AGmath.MG
keywords alexandrov-fenchelconvexgeometryreverseddemaillyfunctionsgeneralizeshigher
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We prove the reversed Alexandrov-Fenchel inequality for mixed Monge-Amp\`ere 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 R. 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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  1. Jumping numbers of analytic multiplier ideals (with an appendix by S\'ebastien Boucksom)

    math.AG 2019-08 conditional novelty 7.0 of 10

    For toric plurisubharmonic functions in dimension 2, the cluster points of jumping numbers are exactly the positive integer multiples of 1/x0 and 1/y0, determined by the asymptotes of the Newton convex body.

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