Research On Some Related Problems In The B-M Homomorphisms And Radial B-M Homomorphisms

This paper's reaserch content belongs to the Brunn-Minkowski theory,which has been developed rapidly in the world in the past decade.This paper reaserch mainly geometric inequalities on Brunn-Minkowski theory like the Shephard problem of Blaschke-Minkowski homomorphism,the Busemann-Petty problem of i-radial Blaschke-Minkowski homomorphism,In addition,we have also studied the Lp-dual mixed geometric surface area and Shephard problem for general Lp-centroid body.In this paper,based on the basic concepts of Brunn-Minkowski theory,Lp-Brunn-Minkowski theory and its dual theory,we solve these problems by means of integral transformation and analytic inequality theory.The concrete results can be stated as follows:1)Based on the study of the concept of Blaschke-Minkowski homomorphism and its Shephard problem,we obtain the asymmetric negative solutions of Shephard type problem of Blaschke-Minkowski homomorphism with volume and affine surface areas respectively.2)Based on the study of the concept of i-radial Blaschke-Minkowski homomorphism and its Busemann-Petty problem,we obtain the affirmative and negative solutions of Busemann-Petty type problem of i-radial Blaschke-Minkowski homomorphism.3)Based on the definition of a new dual geometric surface area given by Wan and Wang,we introduce the concept of Lp-dual mixed geometric surface area and related geometric inequalities are established.4)Based on the study of the Shephard problem of Lp-centroid,we study the positive and negative solutions of the Shephard problem of general Lp-centroid body,and establish the condition of the negative solution of the Shephard problem of the Lpcentroid body,which is extended from the symmetry of origin to the general situation.