The Research Of Dual Blaschke Addition And Related Problems

In this paper,we mainly studied the properties of radial Blaschke addition and related problems.As is known to all,there are different operations between convex bodies in convex geometry and each operation has the corresponding duality concept,for example.the dual of Minkowski addition is radial addition,the dual of Lp addition is radial Lp addition and the dual of Orlicz addition is radial Orlicz addition.Blaschke addition also is one of the important operators in convex geometry.Let K,L be convex bodies on Rn,Blaschke addition "K#L" of K and L is the convex body with the center of mass at the origin and is defined by the following equation S(K#L,·)=S((K,·)+S(L,·).where S(K,·)represents the surface area measure of convex body K.In this paper,the main purpose is to studied the characteristic properties of the dual Blaschke addition and our results is the duality of Gardner's[18]The first chapter briefly introduces the background of source of problem,the research status and the structure arrangement of this paper.The second chapter will briefly introduce the concepts and symbols in convex geometry and the characteristic properties of several common operators,such as:Minkowski addition and radial addition,Lp addition and radial Lp addition,and Orliz addition and dual Orlicz additionIn the third chapter,we will mainly study the dual of m-Blaschke addition about star sets,give the definition of this operator and prove its characteristic properties.This is,given K,L what are the star sets,we define the radial Blaschke addition "KL",which is star set of symmetric about the origin,and makes the equation Vn-1((KL)? u?)=Vn1(Kn u?)+Vn-1(L ? u?),where u?Sn-1.Using the characteristic properties of Minkowski addition,the properties of volume of star set,and the definition of radial Blaschke addition,m-radial Blaschke addition characteristic properties are proved.The fourth chapter mainly studies the concept of intersection body about star set and its correlation inequalities,the ralationship with the m-radial Blaschke addition.We define k-intersection body IK what is a symmetric star set with the center of mass at the origin and make the equation Vk(IkK ? En-k?)=2Vn-k(K ? En-k?),where K is a star set with symmetry about the origin,n—k dimensional subspace En-k ?g(n,n-k).In the process of proof,we used the Minkowski inequality and the polar formula of volume,etc.