The Researches On Some Geometric Inequalities In The Orlicz-Brunn-Minkowski Theory

The research of this dissertation belongs to the field of Orlicz-Brunn-Minkowski theory,which involves the(q,?)-dual Orlicz mixed affine surface areas,asymmetric Orlicz radial body,new dual Orlicz mixed volume,Lp dual mixed geominimal surface areas for multiple star bodies and the Lp-Busemann-Petty problem.The main research methods applied in this paper include integral transforms,analytic inequality and basic concepts contained in the Brunn-Minkowski theory,the Orlicz-Brunn-Minkowski theory and their respective dual theories.The major work of this dissertation can be described as follows:1.Combined with dual mixed volume(called the(q,?)-dual Orlicz mixed volume)proposed by Gardner et al.("General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski problems I”),we introduce(q,?)-dual Orlicz mixed affine surface areas which imply the existing affine surface area.Besides,we also establish product inequality,cyclic inequality and Brunn-Minkowski inequalities.2.Based on the Orlicz radial combination proposed by Baocheng Zhu et al.,we introduce asymmetric Orlicz radial bodies and research several properties and related inequalities.3.Inspired by the work of Lujun Guo et al.("Orlicz extensions of Brunn-Minkowski theory"),we propose new dual Orlicz mixed volume and compute the corresponding variational formula.Further,we give dual Orlicz mixed Minkowski inequality,dual Orlicz mixed Brunn-Minkowski inequality and demonstrate the equivalence between them.4.By defining Lp dual mixed quermassintegrals for multiple star bodies,we consider Lp dual mixed geominimal surface areas and give monotonic inequality and Brunn-Minkowski inequality.5.Associated with Haberl and Ludwig's Lp intersection bodies,we give two negative solutions of Lp-Busemann-Petty Problem.