Research On Three Inequalities In Convex Geometry

The research content of this thesis belongs to convex geometry analysis which have been developing rapidly both home and abroad in recent years.This thesis mainly uses the basic theoretical knowledge of Lp-Brunn-Minkowski theory and geometric analysis to research the following konds of problems:some problems of log-Minkowski type inequalities,volume inequalities of a new class of Sine ellipsoid and An inequality associated with two simplices.Studies on some problems of log-Minkowski type inequalities.Firstly,according to the definition of dual quermassintegral Wi(L)given by Lutwak and the concept of Lp-dual mixed quermassintegral measure dwp,iρ(L,K)given by Weidong Wang and Chao Li,several kinds of log-Minkowski type inequalities are obtained.Secondly,we give two notation about log-Minkowski inequality.On the study of volume inequalities of a new class of Sine ellipsoid.Firstly,a new definition of mixed ellipsoid Λ-2(K,L)is given,and some volume inequlities of it and other ellipsoids are given.Then,two stars T-p(K2,…,Kn,L)and Λ-p(K,L)are defined,and their affine properties and relations are obtain.Finally,the volume inequality between the mixed ellipsoidΛ-2(K,L)and the convex body K is obtained.An inequality associated with two simplices.First,using the method of permutation and combination,the relationship between the length and volume and the area of a side triangle and volume of an n dimensional simplex is discussed.Secondly,a new class of inequality relating twondimensional simplex are established,including some existing special cases.