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 KhovanskiÄ­-Teissier inequality.