Minimal Geometric Surface Area And Dual Orlicz Brunn - Minkowski Theory

In this dissertation, Lp geominimal surface area. Lp mixed geominimal sur-face area, Lp Brunn-Minkowski theory and dual Orlicz-Brunn-Minkowski theory are investigated. Some isoperimetric inequalities for Lp geominimal surface area, analogues of Aleksandrov-Fenchel inequalities for Lp mixed geominimal surface area and dual Orlicz mixed volume inequalities are obtained. All these areas are attracted increased interest by more and more researchers.Geominimal surface area was introduced by Petty in1970’s. Since then it has become apparent that this seminal concept and its general Lp extensions, which are due to Lutwak, serve as bridges connecting affine differential geometry, relative differential geometry and Minkowski geometry. In the second Chapter, a number of Lp affine isoperimetric inequalities for Lp geominimal surface area are established. In particular, a Blaschke-Santalo type inequality is obtained.Chapter three deals with Lp geominimal surface area and its extension to Lp mixed geominimal surface area. It should be noticed that the Lp mixed geomin-imal surface area in this Chapter is refer to p>1, which belongs to Lp Brunn-Minkowski theory. Some inequalities, such as, analogues of Aleksandrov-Fenchel inequalities, Blaschke-Santalo inequalities, and affine isoperimetric inequalities for Lp mixed geominimal surface areas are obtained.In the fourth Chapter, several mixed Lp geominimal surface areas for multiple convex bodies for all-n≠∈R are introduced. The definitions are motivated from an equivalent formula for the mixed p-affine surface area. Some properties, such as the affine invariance. for these mixed Lp geominimal surface areas are proved. Related inequalities, such as, Aleksandrov-Fenchel type inequality, San-talo style inequality, affine isoperimetric inequalities, and cyclic inequalities are established. Moreover, some properties and inequalities for the i-th mixed Lp geominimal surface areas for two convex bodies are also studied.A dual Orlicz-Brunn-Minkowski theory is presented in Chapter five. An Orlicz radial sum and dual Orlicz mixed volumes are introduced. The dual Orlicz-Minkowski inequality and the dual Orlicz-Brunn-MinkowTski inequality are established. The variational formula for the volume of an Orlicz radial sum is established. The equivalence between the dual Orlicz-Minkowski inequality and the dual Orlicz-Brunn-Minkowski inequality is demonstrated. Finally. Orlicz in-tersection bodies are defined and the Orlicz-Busemann-Petty problem is posted.