On The Geometric Inequality Of Gauss Curvature Stars And Related Issues

The isoperimetric inequality is one of the most known inequalities in math-ematics.It states that the ball has maximum volume among all domain of the fixed surface area.The differential affine isoperimetric inequality is a very impor-tant inequality in affine geometry,which characterizes the relationship between the differential affine surface area and volume of smooth convex bodies.Inspired by the concept of affine surface area,we construct a new class of star body and investigate the isoperimetric problem.In chapter 3,we construct a new star body associated with a given smooth convex body,which is called the Gauss curvature star body.The isoperimetric inequality and some other geometric inequalities for Gauss curvature star body are established.Via the isoperimetric inequality and other geometric inequalities obtained,we give some reverse Bonnesen type inequalities.The Minkowski inequality and the Brunn-Minkowski inequality are very im-portant inequalities in convex geometric analysis,which are the extension of isoperimetric inequality.In chapter 3,we generalize the results of chapter 3 and define the mixed Gauss curvature star body.We obtain the Minkowski inequality and Brunn-Minkowski inequality for the mixed Gauss curvature star body.We prove the equivalence of these inequalities.We give the inverse Bonnesen-type Minkowski inequality in Rn.In the Euclidean plane R2,the Wullf isoperimetric inequality is a natural gen-eralization of the classical isoperimetric inequality.In chapter 5,via the Poincare lemma,we establish some Bonnesen type Wulff isoperimetric inequalities and re-verse Bonnesen type Wulff isoperimetric inequalities.Those inequalities obtained are extensions of known Bonnesen type inequalities and reverse Bonnesen type inequalities.