Higher Lelong numbers and convex geometry

TL;DR

This paper establishes a reversed Alexandrov-Fenchel inequality for mixed Monge-Ampère masses, providing new insights into convex geometry and confirming a conjecture on Lelong number convergence for toric functions.

Contribution

It proves a reversed inequality for mixed Monge-Ampère masses and applies it to convex geometry, also confirming a conjecture on Lelong number convergence for toric plurisubharmonic functions.

Findings

Proved reversed Alexandrov-Fenchel inequality for mixed Monge-Ampère masses.

Provided a complex analytic proof for reversed inequalities in convex geometry.

Confirmed Demailly's conjecture on higher Lelong number convergence for toric functions.

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.