Inequalities And Extremum Problems For Convex Bodies And Star Bodies

The development survey and main research directions of convex geometry are presented in the preface. This Ph.D. dissertation research the inequalities and extremum properties for convex bodies, star bodies and some specific convex bodies such as simpli-cies and parallelotopes. The research works of this thesis consists of two parts.In the first aspect, some inequalities and extremum properties are established by applying the asymptotic theory, local theory and integral transforms. The Petty-Schneider problem has attracted the attention of those working in convex geometry. Chapter 2 extend Petty-Schneider theorem for projection bodies (zonoids) to quermassintegrals. Two inequalities for sections of centered bodies are given, which are motivated by the so-called hyperplane conjecture for convex bodies. The classical Brunn-Minkowski's inequality is the heart of convex bodies theory. The Brunn-Minkowski's inequality for mixed projection bodies was obtained by Lutwak. We find the Brunn-Minkowski's inequality for the polar of mixed projection bodies. The dual Brunn-Minkowski theory earns its place as an essential tool in geometric tomography. In Chapter 3, the inequalities about the dual quermassintegrals of star bodies in Rn are established, which are analogue not only to Marcus-Lopes's inequality and Bergstrom's inequality for elementary symmetric functions of positive reals, but also to Fan Ky's inequality for determinant. On the other hand, the dual mixed Quermassintegrals and the dual mixed p- Quermassintegrals are introduced. We generalize the dual Brunn-Minkowski Theory.The other part of the research work is presented in Chapter 4, Chapter 5 and Chapter 6. We find the inequalities and extremum properties for specific convex bodies such as simplicies and parallelotopes by employing the exterior differential methods and algebraic means. The theory of mixed volumes provides a unified treatment of various important metric quantities in geometry, such as volume, surface area and mean width. Chapter 4 introduce the concept of the mixed volume of two finite vector sets in Rn, which can be regard as the discrete form of mixed volume of two convex bodies. An new and powerful inequality associating with the mixed volume of two finite vector sets is obtained. The Cayley-Menger determinant has proved extraordinarily useful tool in dealing with some inequalities for finite points sets. We introduce the mixed Cayley-Menger determinantand obtain the formula for volume product of two simplices which contains a lot of papers on the properties of a pair of triangles (simplices). Meanwhile, the relation concerning the determinant of mixed distance matrix and the circum-radius of two n-simplices is given. Besides, employing new and simple methods, some well-known results of simplices and Hadamard inequality are reproved. In Chapter 5, by applying an analytic inequality and the moment of inertia inequality in Rn, we generalize the Klamkin's inequality to several dimensions and establish some inequalities for the volume, facet areas and distances between any point of Rn and vertices of an n-simplex. To obtain the analytic expression for the mid-facet area of a n-dimensional simplex, the exterior differential method is applied in Chapter 6. Furthermore, some properties of the mid-facets of a simplex analogous to median lines of a triangle (such as for all mid-facets of a simplex, there exists another simplex such that its edge-lengths equal to these mid-facets area respectively, and all of the mid-facets of a simplex have a common point) are confirmed. In the end , by using the analytic expression, a number of inequalities which combine edge-lengths, circum-radius and median line with the mid-facet area for a simplex are established.