A remark on the Alexandrov–Fenchel inequality

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 L2-estimate, which also implies a non-compact version of the Khovanskiĭ–Teissier inequality