The Alexandrov-Fenchel type inequalities, revisited

Various Alexandrov-Fenchel type inequalities have appeared andplayed important roles in convex geometry, matrix theory and com-plex algebraic geometry. It has been noticed for some time thatthey share some striking analogies and have intimate relationships.The purpose of this article is to shed new light on this by compara-tively investigating them in several aspects.The principal resultinthis article is a complete solution to the equality characterizationproblem of various Alexandrov-Fenchel type inequalities for inter-section numbers of nef and big classes on compact K ahler man-ifolds, extending some earlier related results. In addition to thiscentral result, we also give a geometric proof of the complex ver-sion of the Alexandrov-Fenchel inequality for mixed discriminantsand a determinantal generalization of various Alexandrov-Fenchel type in equalities.