Dual Affine Surface Area And Dual Mixed Geominimal Surface Area In The Orlicz Space

This article builds upon the Orlicz Brunn-Minkowski theory,studies the nature of convex body and star body in Orlicz space.On the one hand,Orlicz dual affine surface area for star body are studied.For another part,we explor the natures and inequalities of Orlicz dual geominimal surface area and Orlicz dual mixed geominimal surface area.For research of Orlicz dual affine surface area,first,on the basis of the concept of Orlicz mixed volume put forward by predecessors,we extend Lp-dual affine sur-face area from the Lp space to the Orlicz space,naturally,we get a new notion that is Orlicz dual affine surface area.Then,combining with dual Minkowski inequali-ty and the Blachke-Santalo inequality,we build affine isoperimetric inequality and Blachke-Santalo inequality of Orlicz dual affine surface area.At last,we discuss its monotonicity inequality.In the classical Lp-Brunn-Minkowski theory,Zhu Baocheng proposes the first concept of Lp-mixed geominimal surface area,and some natures and inequalities has proposed.Later,Ma Tongyi defines Orlicz dual geominimal surface area in the Orlicz space.Combining with the research results above,we define the integral form of Orlicz dual geominimal surface area.On this basis,we get two nice inequalities,isoperimetric inequality and Blachke-Santalo inequality.Besides,we define Orlicz dual mixed geonimimal surface area and get Alexandrov-Fenchel type inequality,Blachke-Santalo inequality.Finally,we explore the analogous cyclic inequality and Minkowski inequality of dual ith Orlicz mixed geominimal surface area.