Study Of Extremum Problems For Convex Bodies In Geometric Space

Convex geometry is an important branch of modern geometry, convex bodies and star bodies are the main research objects and Lp-Brunn-Minkowski theory is kernel of convex body theory. This paper, firstly, uses basic knowledge and ap-proaches of Lp-Brunn-Minkowski theory to consider some basic problems on this theory. Meanwhile, by applying Orlicz-Brunn-Minkowski theory, several problems are enlarged to Orlicz space, and get new results in classical Brunn-Minkowski the-ory. Finally, Using algebraic method, the generalized metric equations are obtained in geometric space. In the basis of this, we introduce some extremum problems of two simplexes in geometric space.This article can be stated as follows:(1) In 1996, Lutwak first explicitly introduced the concept of Lp-mixed cur-vature image. Recently, the notion of ith Lp-mixed curvature image was given by others. In this article, we use the Brunn-Minkowski-Fiery theory, by studying the concept and properties of ith Lp-mixed curvature image, and establish some in-equalities involving the ith Lp-curvature image and the quermassintegrals (or dual quermassintegrals).(2) Since Lutwak, Yang and Zhang, respectively, introduced the notion of orlicz projection and orlicz centroid, the Orlicz-Brunn-Minkowski theory and dual of the theory was gradually formed, and developed rapidly. In this paper, the notion of dual affine quermassintegrals in the dual Brunn-Minkowski theory is extended to that of dual orlicz mixed affine quermassintegrals in the dual Orlicz-Brunn-Minkowski the-ory. The analogs of the Minkowski inequality and the Brunn-Minkowski inequality are established for this new dual orlicz mixed affine quermassintegrals.(3) As the generalization of the well-known Cavlev-Menger algebra, the metric equations has become the one of the most major study objects by the majorities of basic work of Yang and Zhang. In this article, we give a concept of bifundamen-tal figurates in spherical space. Using algebraic method, the metric equations in spherical space, given by Yang Lu and Zhang Jingzhong, are generalized to bifun-damental figurates, and obtain the generalized metric equations. As their primary applications, some formula involving two simplexes are given in spherical space.