  Normal complexes are orthogonal truncations of polyhedral fans. In this
paper, we develop the study of mixed volumes for normal complexes. Our main
result is a sufficiency condition that ensures when the mixed volumes of normal
complexes associated to a given fan satisfy the Alexandrov-Fenchel
inequalities. By specializing to Bergman fans of matroids, we give a new proof
of the Heron-Rota-Welsh Conjecture as a consequence of the Alexandrov-Fenchel
inequalities for normal complexes.
