Affine Isoperimetric Inequalities And The Orlicz Brunn-Minkowski Theory

The researches of this thesis belong to the theory of convex geometric anal-ysis, and devoted to the study of amne isoperimetric inequalities and the Orlicz Brunn-Minkowski theory. Brunn-Minkowski theory is the core of the convex ge-ometric analysis. It originated with the study of the relationship between the volumes of two convex bodies and their Minkowski sum. The celebrated Brunn-Minkowski inequality emerged from this study, and it soon become the corner-stone of the Brunn-Minkowski theory. Amne isoperimetric inequalities focus on the extremal problems associated with amne quantities of convex bodies. It is an important ingredient of the convex geometric analysis. Classical amne quantities of convex bodies include amne surface area, the volume of projection body, the volume of centroid body, and so on. They have been found spread uses in the fields of information theory, and analysis. Classical isoperimetric inequality character-izes the Euclidean ball, while the amne isoperimetric inequalities can characterize many other geometric objects, such as ellipsoids, cubes, and simplices.In chapter 2, we study the framework of the Orlicz Brunn-Minkowski theory. This theory originated with the work of Lutwak, Yang, and Zhang in 2010. In this chapter, we first introduce the Orlicz addition of convex bodies containing the origin in their interiors, and then extend the Lp Brunn-Minkowski inequality to the Orlicz Brunn-Minkowski inequality. Furthermore, we extend the Lp Minkows-ki mixed volume inequality to the Orlicz mixed volume inequality by using the Orlicz Brunn-Minkowski inequality. These results generalize the corresponding results in Lp Brunn-Minkowski theory.In chapter 3, we obtain the amne isoperimetric inequalities of Lp mean zonoids. The notion of mean zonoid is denned by Zhang [116], and in that paper, he firstly proved the famous reverse Petty projection inequality. He also proved the affine inequality for the mean zonoid by using an integral geometric approach. It seems that we could not get a proof using the same method as there. By utilizing the Steiner symmetrization, we get a proof of the affine isoperimetric inequalities of this new bodies.At the beginning of the final chapter, we study Dar’s conjecture. It is pose by the Israel mathematician Dar in 1999, and it will be a stronger version of the Brunn-Minkowski inequality. In 2011, Campi, Gardner, and Gronchi [14, Page 1208] pointed out that Dar’s conjecture "seems to be open even for planar o-symmetric bodies". We prove that Dar’s conjecture is correct for all planar convex bodies, and we also obtain the equality conditions. For planar o-symmetric convex bodies, the log-Brunn-Minkowski inequality was established by Boroczky, Lutwak, Yang, and Zhang in 2012. It is stronger than the classical Brunn-Minkowski inequality, for planar o-symmetric convex bodies. They asked if there is a general version of this inequality. Fortunately, the solution of Dar’s conjecture, especially, the definition of "dilation position", inspires us to obtain a general version of the log-Brunn-Minkowski inequality. As expected, this inequality implies the classical Brunn-Minkowski inequality for all planar convex bodies.