On Several Problems Of L_p-Space Theory

The researches of this thesis belongs to the theory of convex geometric func-tional analysis, and devotes to the study of functional inequalities and extremelyproblems of the L_p-Brunn-Minkowski theory. Our main works are to researchthe theories of convex bodies, some inequalities and extreme properties of geome-try bodies by applying the basic notions, basic theories and integral transforms ofL_p-Brunn-Minkowski theory.The purpose of chapter 2 is to study comprehensively the Aleksandrov bodywhich is an essential matter in the Brunn-Minkowski theory. The notion of Alek-sandrov body was firstly introduced to solve Minkowski problem in 1930s. TheAleksandrov body establishes the relationship between the convex body containingthe origin and the positive continuous functions, and characterizes the convex bodyby means of the positive continuous functions. We firstly introduce a new geomet-ric body which is called p-Aleksandrov body. We not only study the qualities,and establish the Brunn-Minkowski inequality and the L_p-Minkowski inequalityfor the Aleksandrov bodies associated with positive continuous functions, but alsoget some functions equivalent conditions by studying the positive continuous func-tions characterized the Aleksandrov body. We establish the L_p-Brunn-Minkowskiinequality for the p-Aleksandrov body and the Aleksandrov body. Meanwhile,we get the L_p-Kneser-Su¨ss inequality for the Aleksandrov body associated withfunction f. As the applications of these inequalities, we prove the convergent resultabout the Aleksandrov bodies associated with positive continuous functions.The purpose of chapter 3 is to study a special case of Mahler volume for theclass of symmetric convex bodies in R~3. One of the main open questions is theproblem of finding a sharp lower estimate for the Mahler volume of a convex bodyK in the theory of convex geometric functional analysis. We give the definitions of a generalized cylinder, a body of revolution and a generalized bicone in R~3. Weprove that among the generalized cylinders, a cube has the minimal Mahler volumeand a cylinder has the maximal Mahler volume by the basic theory of L_p-space.Then, we get the Mahler volume of the body of revolution obtained by rotating theunit disk in R~2.The purposes of chapter 4 and chapter 5 are to study the basic theory of thedual L_p-Brunn-Minkowski theory. In chapter 4, we study the special case of p = 0in the L_p-theory. We define the radial combination of the dual 0-sum, from whichwe obtain a new star body, and then study some related properties. Moreover, weestablish the integral representation of L_p-dual mixed volume, study its qualities,prove its bounded theory and convergence theory, and get the Minkowski inequality.In chapter 5, we give a new definition of L_p-dual mixed quermassintegral.We not only study its qualities and integral representation, but also establish theMinkowski inequality and the Brunn-Minkowski inequality of L_p-dual mixed quer-massintegrals. Based on the notions of L_p-affine surface area and L_p-dual affinesurface area, we give a new definition of L_p-dual mixed affine surface area byintroducing the notion of i-th L_p-mixed affine surface area. Then, we extendsome Lutwak's results for the mixed affine surface area, and establish some relatedinequalities.