Research On Some Problems Of Convex Body In Lp-space

The research content of this article belongs to the Lp-Brunn-Minkowski theory in theconvex geometric analysis, which mainly devote to study the extreme value problems andthe geometric inequalities of some geometric bodies and geometric measure on theLp-Brunn-Minkowski theory. we chiefly use basic concepts, basic methods and integraltransforms of Lp-Brunn-Minkowski theory to study some notions such as Lp-dual affinesurface area、Lp-dual geometric surface area、Lp-projection bodies、Lp-centroid bodies、Lp-mixed curvature images as well as width integrals and chord integrals. The following isour main results:The chapter1of this article states the background of the Lp-Brunn-Minkowski theoryand main results.In chapter2, we mainly study Lp-dual affine surface areas and Lp-dual geometricsurface areas of the Lp-Brunn-Minkowski theory. Based on the notions of Lp-geominimal surface area and Lp-dual geominimal surface area, we researcher someinequalities related to them, such as the dual version of affine isoperimetric inequalities ofLp-dual affine surface area、affine isoperimetric inequalities of Lp-dual affine surface arearelated to Lp-curvature images and another form of Brunn-Minkowski type inequalitiesof Lp-dual affine surface area. Besides, the affirmative form of Shephard type problems ofLp-affine surface area version is established. We also introduce the notion of Lp-harmonicBlaschke bodies, which obtains the nagative form of Shephard type problems of Lp-affinesurface area version. Finally, we introduce the notions of Lp-dual mixed geominimalsurface area, and study its property, Blaschke-Santalo inequality and Brunn-Minkowskiinequality, and so on.In chapter3, based on the notions of Lp-mixed centroid bodies and Lp-mixedcurvature images, the first aim of this chapter is to establish another form of the affirmativeform of Shephard type problems of Lp-mixed centroid bodies, and also give its negativeform. Further, an extended version of Funk’s section theorem is obtained. Finally, weestablish two monotone inequalities of dual quermassintegrals related to Lp-mixed centroidbodies. Another aim is to establish the monotone inequalities and cyclic inequality of dualquermassintegrals related to Lp-mixed curvature images.In chapter4, based on the notions of Blaschke-Minkowski homomorphisms and radialBlaschke-Minkowski homomorphisms in valuation theory of convex bodies, this chapterestablishs Minkowski, Brunn-Minkowski and Aleksandrov-Fenchel type inequalities ofdifference of quermassintegral and dual quermassintegrals of mixed Blaschke-Minkowskiand mixed radial Blaschke-Minkowski homomorphisms as well as Minkowski and Brunn-Minkowski type inequalities of difference of quermassintegral of mixed Blaschke-Minko-wski homomorphisms. Moreover, for the Blaschke-Minkowski homomorphisms, theaffirmative and negative forms of Shephard type problems of Lp-affine surface area version are given. Finally, we also obtain several Brunn-Minkowski type inequalities ofwidth integrals related to Blaschke-Minkowski homomorphisms.In chapter5, general Lp-projection bodies are introduced by Ludwig. We speciallydiscuss its Petty affine projection inequalities； Based on such notions as centroid bodies、Lp-centroid bodies、Lp-mixed centroid bodies，we introduce general Lp-centroid bodiesmotivated by the recent work on general Lp-affine isoperimetric inequality by Haberl andSchuster，and establish two monotone inequalities and Brunn-Minkowski inequality. Inparticular, the extremal values of dual quermassintegrals of the polars of generalLp-centroid bodies are provided; general Lp-centroid bodies are introduced, and studysome of its property as well as obtain the extremal values concerning the volume andLp-dual geominimal surface area of this new notion; We define general mixed widthintegrals of convex bodies and study some of its properties. Some inequalities such asaffine isoperimetric inequalities、Aleksandrov-Fenchel type inequality、Blaschke-Santalóinequality and cyclic inequality related to them are established. In particular, We also givethe more general Brunn-Minkowski type inequalities about it; Similar to the definition ofgeneral mixed width integrals of convex bodies, we finally also define the notion ofgeneral mixed chord integrals of star bodies and obtain some similar ones with the resultsof general mixed width integrals of convex bodies.