Research On Geometric Inequalities In Asymmetric Lp-Brunn-Minkowski Theory

This paper is devoted to the study of some geometric inequalities in asymmetric Lp-Brunn-Minkowski theory.Asymmetric Lp-Brunn-Minkowski theory belongs to the field of convex geometric analysis.In recent years,this theory has developed rapidly at home and abroad.In this paper,we mainly use Brunn-Minkowski theory,Lp-Brunn-Minkowski theory and the basic knowledge and concepts of dual theory,integral transformation method and analytic inequality theory to study some general geometry bodies,including general Lp-centroid bodies,general Lp-intersection bodies,general Lp-projection bodies and research on(p,q)-mixed volume,(p,q)-mixed geominimal surface area,Lp radial Blaschke-Minkowski homomorphism and the general chord-integral of index i.The innovations of this paper are as follows:1.Based on the definition of general Lp-centroid bodies,firstly,we study the Shephard type problem,which is one of the most influential geometric problems in Lp-Brunn-Minkowski theory.Then,we combine Lp dual affine surface area with general Lp-centroid bodies to discuss the extremum values,Lp-Brunn-Minkowski type inequality and other related inequalities for the polar.2.Based on the general Lp-intersection bodies introduced by Professor Wang Weidong and Li Yanan in 2015,we study its extremum problem and establish Lp-Brunn-Minkowski type inequality and other related inequalities.3.In 2018,Lutwak,Deane Yang and Zhang Gaoyong gave the definition of(p,q)-mixed volume.Combine with the general Lp-projection bodies and the general Lp-centroid bodies,we study their related Brunn-Minkowski type inequalities.Later,based on this definition,Feng Yibin and Professor He Binwu gave the definition of(p,q)-mixed geominimal surface area.We study its monotone inequality,and establish Brunn-Minkowski type inequality and other related inequalities firstly.Then,we combine with Lp radial Blaschke-Minkowski homomorphism,and discuss the related Brunn-Minkowski type inequality,cyclic inequality and monotone inequality.4.Based on the research of general mixed chord-integral and the concept of radial Blaschke-Minkowski homomorphism,we give the Brunn-Minkowski type inequality and difference inequality about i-th general chord-integral.