Inequalities And Extremum Problems For Mixed Bodies

This article belong to the domain, is a high-speed developing geometry branch on the decade of late, of the Lp- Brunn-Minkowski theory (or called Brunn-Minkowski-Firey theory).Our main works are to research the theories of convex bodies,some inequalities and extremum properties of geometry bodies by applying the basic notions, basic theories and integral transforms of the Lp- Brunn- Minkowski theory.In the first aspect,we research the inequalities and extremum properties of some geometry bodies containing the new geometry body Ip,iK and mixed centroid body.The other parts,the Neuberg-Pedoe inequality of two triangles for k-dimensions are re-searched.As the introductions, in Chapter 1, the development survey and main research directions of convex geometry are presented in the preface.In Chapter 2, by using Brunn-Minkowski-Firey mixed volume theory and dual mixed volume theory,associated with the new geometric body and dual mixed volume,some dual Brunn-Minkowski inequalities and their isolate forms are established for the new geometric body about Lp radial linear combina-tion.In addition,the monotone inequalites for volume of the new geometric body are obtained.In Chapter 3, by using Brunn-Minkowski-Firey mixed volume theory and dual mixed volume theory,associated with the mixed centroid body and dual mixed volume,some Brunn-Minkowski inequalities are established for the mixed centroid body about Lp radial linear combination.In addition,the in-equalites for volume of the mixed centroid body are obtained.In the last Chapter, some geometric inequalities for the volumes of two n-simplexes and their k-subsimplexes are established. The results are gen-eralization to several dimensions of the well-known Neuberg-Pedoe inequality of two triangles.