Blaschke-Minkowski Homomorphisms And The General L_p Blaschke Body

This dissertation deals with topics in valuations theory and Orlicz-Brunn-Minkowski theory.Valuations theory is the core of the convex geometric analysis.Orlicz-Brunn-Minkowski theory originated with the work of Lutwak,Yang,and Zhang in 2010.This dissertation is devoted to Orlicz-Brunn-Minkowski inequal-ities for Blaschke-Minkowski homomorphisms,Lp-Brunn-Minkowski inequalities for Blaschke-Minkowski homomorphisms and related extremal problems.In chapter 2,we establish the Orlicz-Brunn-Minkowski inequalities for Blaschke-Minkowski homomorphisms.The results extend the Lp Brunn-Minkowski for Blaschke-Minkowski which were established by Wang[115].Schuster[101]intro-duced Blaschke-Minkowski homomorphisms and radial Blaschke-Minkowski ho-momorphisms which are more general than the well known projection body oper-ators and intersection bodies,respectively.In particular,Schuster has established the Brunn-Minkowski type inequalities for Blaschke-Minkowski homomorphisms.In chapter 3,we establish the Lp Brunn-Minkowski inequalities for Lp Blaschke Minkowski homomorphisms.W.Wang[114]introduced the concept of the Lp Blaschke-Minkowski homomorphisms which extended the definition of projec-tion body operator.W.Wang[113]also established the definition of Lp radial Minkowski homomorphisms which extended the concept of intersection body op-erator.Our results extended the inequalities established by F.Lu and G.Leng.In chapter 4,based on the Lp Minkowski problem for p>1,we define the general Lp Blaschke addition and general Lp Blaschke bodies,and obtain the extremal values of their volume and Lp affine surface area.Further,as the applications,we study two negative forms of the Shephard problems.In chapter 5,we prove several new inequalities for volume sums for polars of mixed complex projection bodies.Complex version of the Busemann-Petty problem was solved by A.Koldobsky,H.Konig and M.Zymonopoulou[55].J.Abardia and A.Bernig[3]introduced the notion of complex projection body and obtained the characterization theorem of the complex projection body oper-ator.They also established the general Minkowski inequality,Brunn-Minkowski inequality and Aleksandrov-Fenchel inequality for mixed complex projection bod-ies.Liu[64]established general Minkowski inequality,Brunn-Minkowski inequal-ity and Aleksandrov-Fenchel inequality for polars of mixed complex projection bodies.We establish the Minkowski,Brunn-Minkowski and Aleksandrov-Fenchel type inequalities for volume sums of polars of mixed complex projection bodies.