Higher Lelong numbers and convex geometry

2018/03/21 by Kim, Dano, Rashkovskii, Alexander · 1 citation
#14F18 #32U05 #32U25 #52A39 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1803.07948

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 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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