Extremal Problems Of Orlicz Mixed Intersection Bodies And Orlicz Mixed Geominimal Surface Area

First,this paper introduce Lp-dual mixed quermassintegral quotient function on the basis of Lp-dual mixed quermassintegrals.Then,we use basic knowledge and approaches of Orlicz Brunn-Minkowski theory to make a further study of geometry for intersection bodies and geominimal surface area of bodies in Orlicz space.Our main results can be stated as follows:(1)Based on the concept of Lp-dual mixed quermassintegrals by Wang Weidong et al.,we introduce the concept of LP-dual mixed quermassintegral quotient function of star bodies.By applying the integral methods and combining with the radial Minkowski linear combination,the radial Blaschke linear combination and the harmonic Blaschke linear combination,some Brunn-Minkowski type inequalities for Lp-dual mixed quermassintegral quotient function are established.(2)On the basis of Orlicz intersection bodies by Ma Tongyi et al.,we investigate the Orlicz mixed intersection bodies and its properties.By using geometric analysis,we introduce the concept of Orlicz mixed intersection bodies.And we obtain the continuity and affine invariant property of the Orlicz mixed intersection bodies operator.By applying the integral methods and Steiner symmetrization,the affine isoperimetric inequality for Orlicz mixed intersection bodies is established.(3)In 2010,Lutwak,Yang,Zhang introduced Orlicz centroid bodies and Orlicz pro-jection bodies,which laid the foundation of convex geometry and analysis in the Orlicz Brunn-Minkowski theory.We deal with the Orlicz geominimal surface area on the basis of predecessors and give an integral representation by the Orlicz-Petty body.Further-more,we introduced the notion of Orlicz mixed geominimal surface area and ith Orlicz mixed geominimal surface area.Then some related inequalities are established,includ-ing Alexandrov-Fenchel type inequality,analogous cyclic inequality,Blaschke-Santalo type inequality,and affine isoperimetric inequality.