Affine Isoperimetric Inequality Of Orlicz Projection Mean Ellipsoid

This paper belongs to Brunn-Minkowski theory.In the framework of Orlicz Brunn-Minkowski theory,we study the evolution of the mean ellipsoid system discovered by Zou Du and Xiong Ge in Orlicz space,and the affine extremum problem under affine invariants in Orlicz space.These studies are widely used in the analysis of inequalities.Firstly,in the framework of Orlicz Brunn-Minkowski theory,using the mixed affine homogeneous integral,the integral affine surface area in Orlicz space is derived,and the first-order variational formula of the mixed integral affine surface area is obtained,The affine invariance of integral affine surface area of Orlicz mixture of convex bodies is introduced.Based on the affine invariance of mixed affine mean integral in Grassmann popular technology,the affine invariance of integral affine surface area of Orlicz mixture is obtained.Secondly,the concept of projection mean ellipsoid introduced by Zou Du and Xiong Ge in Euclidean spaceR~n is extended to Orlicz Brunn-Minkowski theory framework,and the Orlicz projection mean ellipsoid system is obtained.The continuity of Orlicz projection mean ellipsoid is proved.Finally,the Orlicz Brunn-Minkowski inequality is established for the integral affine surface area of Orlicz mixed convex body,and the corresponding affine equal cycle inequality is established by combining the average ellipsoid of Orlicz projection,and the affine isoperiodic inequality of the average ellipsoid of Orlicz projection is obtained.