Research On Some Related Problems Of Blaschke-Minkowski Homomorphism And Its Radial Form In Brunn-Minkowski Theory

The research content of this paper belongs to Brunn-Minkowski theory and LpBrunn-Minkowski theory,which is devoted to study the Busemann-Petty problems for Blaschke-Minkowski homomorphisms,affine surface area inequalities of Lp BlaschkeMinkowski homomorphisms,the Busemann-Petty problems for i-th Lp radial BlaschkeMinkowski homomorphisms,Blaschke-Minkowski homomorphisms in complex vector space and related inequalities,complex intersection bodies and related inequalities,Lp and Lp-dual affine surface inequalities of Lp-curvature image,(p,q)-dual mixed volume and related inequalities.The main research methods and skills used in this paper are integral transforms,some basic concepts which contained in the analytic inequality theory,the Brunn-Minkowski theory,the Lp-Brunn-Minkowski theory and its dual theory.This paper mainly has the following results:1.Associated with the notion of Blaschke-Minkowski homomorphisms,we research a type of Shephard problems for Blaschke-Minkowski homomorphisms.2.Based on the concept of Lp Blaschke-Minkowski homomorphisms,we establish the Brunn-Minkowski type,cyclic type,and monotonic type inequalities for Lp Blaschke-Minkowski homomorphisms.In addition,we also study the Busemann-Petty problems of Lp Blaschke-Minkowski homomorphisms.3.Based on the notions of radial Blaschke-Minkowski homomorphisms and i-th Lp radial Blaschke-Minkowski homomorphisms,we study the Busemann-Petty problems for i-th Lp radial Blaschke-Minkowski homomorphisms.4.Based on the concept of Blaschke-Minkowski homomorphisms defined by Schuster,we define the notion of complex Blaschke-Minkowski homomorphisms,and give the related inequalities.5.Based on the notion of complex intersection bodies,we establish the volume difference fuction for complex intersection bodies,and study the Busemann-Petty problems of complex intersection bodies.6.Based on the concepts of Lp and Lp dual affine surface area,according to the Lp curvature image,we give the related inequalities of Lp curvature image.7.Combined with the definition of(p,q)-dual mixed volume,we establish some inequalities of(p,q)-dual mixed volume.