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Higher Lelong numbers and convex geometry
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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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Jumping numbers of analytic multiplier ideals (with an appendix by S\'ebastien Boucksom)
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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