  Various Alexandrov-Fenchel type inequalities have appeared and played
important roles in convex geometry, matrix theory and complex algebraic
geometry. It has been noticed for some time that they share some striking
analogies and have intimate relationships. The purpose of this article is to
shed new light on this by comparatively investigating them in several aspects.
\emph{The principal result} in this article is a complete solution to the
equality characterization problem of various Alexandrov-Fenchel type
inequalities for intersection numbers of nef and big classes on compact
K\"{a}hler manifolds, extending some earlier related results. In addition to
this central result, we also give a geometric proof of the complex version of
the Alexandrov-Fenchel inequality for mixed discriminants and a determinantal
generalization of various Alexandrov-Fenchel type inequalities.
