Research On Some Related Problems Of The Blaschke-Minkowski Homomorphisms And Radial Forms

The main contents of this paper belong to the fields of Blaschke-Minkowski homomorphism and radial Blaschke-Minkowski homomorphism(Hereafter referred to as B-M homomorphism and radial B-M homomorphism),which have developed rapidly in recent ten years.We mainly use the basic concepts of(Lp)Brunn-Minkowski theory and their dual theory,assignment theory,integral formula,H?lder and Minkowski integral inequality,integral transformation and geometric analysis to study some problems of(Lp)B-M homomorphism,(Lp)radial B-M homomorphism,(p,q)mixed volume and(p,q)mixed affine surface areas.The main results are be stated as follows:1.Based on the concept of B-M homomorphism,the difference inequalities of quermassintegrals and dual quermassintegrals are given.2.Based on the definitions of radial B-M homomorphism and star duality,we establish Brunn-Minkowski inequalities and monotone inequality,besides,we study the related Busemann-Petty problems.3.Based on the concept of Lp-B-M homomorphism,we study the Shephard problem of Lp geometric surface area,and giving the positive and negative solutions.4.Based on Lp and Lp radial B-M homomorphisms,we establish their Brunn-Minkowski type inequalities.5.Based on the concept of Lp radial B-M homomorphism,we establish BrunnMinkowski inequality,monotone inequality and study its Busemann-Petty problems.6.Based on(p,q)mixed volume,we establish some inequalities of Brunn-Minkowski type on B-M homomorphism.7.Based on(p,q)mixed affine surface area,we establish cyclic inequalitiy,monotone inequalities and product inequality.