Title: The Alexandrov-Fenchel type inequalities, revisited,
Abstract: 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
Kähler manifolds, extending earlier results of Boucksom-Favre-Jonsson,
Fu-Xiao and Xiao-Lehmann. Our proof combines a result of Dinh-Nguyên on
Kähler geometry and an idea in convex geometry tracing back to Shephard. In
addition to this central result, we also give a geometric proof of the complex
version of the Alexandrov-Fenchel type inequality for mixed discriminants and a
determinantal type generalization of various Alexandrov-Fenchel type
inequalities.