Properties For Section Bodies And Dual Mixed Volume In L_p-space

This article is mainly about researching the nature of some special bodies in convex geometry. We could get some experience by studying, in the good condition(for example, if the convex bodies and star bodies containing the origin). we could find that there is important relationship among the Lp-dual mixed volume; Lp-dual quermassintegral; quotien star body; width function; Lp-section body and their polar bodies, and with these famous inequalities (for example, the Holder inequality, the Jensen inequality, and the Lp-Busemann-Petty center of mass inequality), a series of graceful and interesting conclusions will be obtained; In this article, the i-rank quermassintegral difference function and Lp-dual quermassintegral sum function are analogied to define two new functions, to go a step further, we discuss their some properties in special conditions.In Chapter1, the development process of convex geometry and research achievements of this article is introduced.In Chapter2, the Lp-dual mixed volume, the Lp-centroid body, and the quotien star bodies’relationship is mainly discussed. At the same time, the factor A is especially researched for the effect of development.In Chapter3, the width function is simply introduced, analogying the i-rank quermassintegral difference function Dωi(K, D) and Lp-dual quermassin-tegral sum function SVp (K, D), two new function Dωp,i(A, B) and SV-p,D (K, L) are respectively defined. To go a step further, some inequalities could be es-tablished when i=2or i=1.In Chapter4, some inequalities on the Lp-section body, the Lp-centroid body and their polar bodies are established with several famous inequalities.