The Researches Of Some Geometric Inequalities In The Brunn-Minkowski Theory

The research content of this thesis belongs to Brunn-Minkowski theory,which is the leading research field of convex geometry analysis in the world.This thesis mainly uses Brunn-Minkowski theory,functional analysis and geometric analysis to research the geometric inequalities,extremum problem and Shephard problem also call Busemann-Petty problem of Lp spaces and complex vector spaces.In Chapter 1,we mainly introduce the development and research trends of BrunnMinkowski theory.In addition,we also introduce our main research results.In Chapter 2,we define the Lp-mixed quermassintegrals probability measure and obtain the log-Minkowski inequality for the Lp-mixed quermassintegrals.As its application,we establish the Lp-mixed affine isoperimetric inequality.In addition,we also consider the dual log-Minkowski inequalities for the Lp-dual mixed quermassintegrals.In Chapter 3,we further study the general Lp geometry bodies and geometry measure in Lp space.In this chapter,we study the Shephard type problem of the general Lp-projection bodies for the Lp-geominimal surface area.Secondly,we further research the property of the general Lp-intersection bodies,meanwhile,the extreme value problem of the general Lp intersection bodies with respect to dual quermassintegrals is obtained.Then,we establish difference types of cyclic inequalities of general Lp-mixed width-integrals and chord-integrals,and obtain the related Minkowski type inequalities.Finally,we define the concept of dual mixed affine surface areas of multiple stars.In addition,we establish the related cyclic inequality,monotonic inequality,product inequality and Brunn-Minkowski inequality.In Chapter 4,By researched the concept of Blaschke-Minkowski homomorphism proposed by Schuster in 2006.We establish some Brunn-Minkowski inequalities for general width-integrals of Blaschke-Minkowski homomorphisms.As applications,inequalities for width-integrals of projection bodies are derived.In Chapter 5,we generalize the concept of mixed brightness integrals to complex vector space and define the concepts of mixed complex brightness integrals and dual mixed complex brightness integrals,meanwhile we establish the related Brunn-Minkowski type inequality,Aleksandrov-Fenchel inequality,cyclic inequality and monotone inequality.As an application,we also establish the difference inequalities of mixed complex brightness integrals and dual mixed complex brightness integrals.Then,Haberl firstly introduced the notion of complex centroid body.Based on this concept,we establish some related inequalities containing Brunn-Minkowski type inequalities and monotonic inequalities.In addition,we also study its Shephard type problem.