A remark on the Alexandrov-Fenchel inequality

Abstract

In this article, we give a complex-geometric proof of the Alexandrov-Fenchel inequality without using toric compactifications. The idea is to use the Legendre transform and develop the Brascamp-Lieb proof of the Prékopa theorem. New ingredients in our proof include an integration of Timorin's mixed Hodge-Riemann bilinear relation and a mixed norm version of Hörmander's $L^2$-estimate, which also implies a non-compact version of the Khovanskiĭ-Teissier inequality.

New version, "on line first" in Journal of Functional Analysis: https://doi.org/10.1016/j.jfa.2018.01.016

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