The Inverse Theorem Of Steiner Symmetrization Of Convex Bodies And The Properties Of L_p Addition

In convex geometry analysis and related fields,Steiner symmetrization is very typical and useful tools in convex body theory.The purpose of Steiner symmetriza-tion proposed by Jakob Steiner is to solve the problem of isoperimetric inequality,but its effect was soon spreaded to other areas.For example,some theorems relat-ing to classical geometry and analysis which need to use Steiner symmetrization.In this paper,some basic knowledge and Steiner symmetrization's typical applications are reviewed.In addition,we mainly researched the converse theorem of Steiner symmetrization of convex body and the other symmetrization on convex body,and some related properties are obtained.Finally,the properties of the Steiner sym-metrization of convex body about the Lp addition are obtained.The structure of this thesis is organized as follows:Chapter One:Preliminaries.In this chapter,the basic concepts and relevant conclusions of Steiner symmetrization and the Lp addition are recapitulated,which will be used throughout this thesis.Chapter Two:The inverse theorems of the Steiner symmetrization on convex body.In this chapter,the inverse theorem of the Steiner symmetrization of the con-vex body and some examples about the properties of the convex body symmetriza-tion are given.Firstly,according to the properties of Steiner symmetrization,such as volume-preserving,convexity-preserving,monotonicity,surface area reduction and so on,a transformation Tu on convex body can be constructed.Secondly,in accor-dance with conditions about Tu and Steiner symmetrization's characterization,it is proved that Tu is Steiner symmetrization and acquire two homologous corollaries.Finally,we discuss some other symmetrization of convex body and give some simple examples.Chapter Three:The Steiner symmetrization of convex body relating to the nature of the Lp addition.We know that there are many properties in the Steiner symmetrization of convex body.The Steiner symmetrization of convex body has inclusion relationships under Minkowski addition,then what are the properties of the Steiner symmetrization under Lp addition?In this chapter,according to the relationship between the addition of Minkowski and Lp,we get some properties relating to the Steiner symmetrization about the Lp addition of the convex body in some specific conditions.