Researches On Cyclic Brunn-Minkowski Inequality And Related Problems

In this paper,we mainly study the contents and derivation forms of the cyclic Brunn Minkowski inequality,and analyze its contents with various research methods.The analysis of its content and research methods has become more and more mature,the research scope of this paper belongs to Brunn-minkowski theory and Lp-Brunn-minkowski theory which have been developing rapidly in recent decades.Based on the Minkowski integral inequality and H?lder integral inequality,the cyclic Brunn Minkowski inequality for dual mixed volume and four other generalized geometry is established.The geometry includes the following four parts: general mixed width integrals,general mixed chord integrals,general Lp-mixed brightness integrals and general Lp-dual mixed brightness integrals.Then,according to the concept and properties of general Lp dual mixed brightness integral,a series of important inequalities are given.The following results are obtained in this paper:1.On the basis of the concepts of general mixed width integrals and general mixed chord integrals given by Yibin Feng and Weidong Wang,we establish the cyclic BrunnMinkowski inequality for these two kinds of geometry,and generalize some of their results on the basis of this inequality.2.Based on the definition of the general Lp-mixed brightness integrals given by Li Yan and Weidong Wang,we give its cyclic Brunn-Minkowski inequality,using the general Lp-dual mixed brightness integrals given by Ping Zhang and Weidong Wang,On the basis of these two inequalities,a series of Brunn Minkowski type inequalities and cyclic inequalities are derived.3.According to the basic definition and properties of dual mixed volumes,the cyclic Brunn-minkowski inequality was established and some meaningful results of Lutwak were derived.4.For the content of the general Lp dual mixed brightness integral,we further study the Lp dual mixed brightness integral,and obtain the Minkowski inequality,monotone inequality,product inequality and quotient inequality of these two different brightness integrals.