  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 Pr\'ekopa
theorem. New ingredients in our proof include an integration of Timorin's mixed
Hodge-Riemann bilinear relation and a mixed norm version of H\"ormander's
$L^2$-estimate, which also implies a non-compact version of the
Khovanski\u{i}-Teissier inequality.
