Higher Lelong numbers and convex geometry

Abstract

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.