Some Class Of Brunn-Minkowski Inequalities

In this thesis, we investigate some inequalities in the Brunn-Minkowski theory, that is, we study the Lp mixed volumes, dual Lp mixed volumes, the affine surface area and some related geometric inequalities. Using Holder inequality, Minkowski inequlity and other inequalities, we obtain reverse dual Lp-Minkowski inequality, reverse dual Lp Brunn-Minkowski inequality, reverse Minkowski type inequalities on affine surface area and reverse Brunn-Minkowski-type inequality. At the same time, we get Lp mixed volume cycle inequality, the stronger dual Lp mixed volume cycle inequality, the relationship between dual Lp-Minkowski inequality and dual Lp Brunn-Minkowski inequality.The classical Brunn-Minkowski theory, the so-called mixed volumes theory, is one of the most important theory in convex geometric analysis. It was born in the dissertation of H. Brunn and the creative work of H. Minkowski. The theory of convex bodies became an independent branch of geometry through hard works of T. Bonnesen, W. Fenchel, A. D. Aleksandrov, W. J. Firey, E. Lutwak, G. Zhang, R. J. Gardner and many other famous mathematicians. The dual Brunn-Minkowski theory, originated from E. Lutwak’s famous paper "Dual mixed volumes" in 1975, is closely related and paralleled to the Brunn-Minkowski theory. From 1980s, there are many important results that enriched the theories of convex bodies and star bod-ies, for instance, following Firey’s Lp linear combination, E. Lutwak have extended the classical Brunn-Minkowski theory to Lp-Brunn-Minkowski theory, which was generalized to Orlicz-Brunn-Minkowski theory in 2010 by E. Lutwak, Yang, Zhang, Garnder, etc.In Chapter 1 and Chapter 2, we introduce the background and some prelimi-naries, respectively.In Chapter 3, we study the dual Lp-Minkowski inequality, dual Lp-Brunn-Minkowski inequality, Beckenbach-Dresher’s inequality and Brunn-Minkowski in-equalities for affine surface area. By a series of analytic inequalities, we get the in-verse of dual Lp-Brunn-Minkowski inequality, Beckenbach-Dresher’s inequality and Brunn-Minkowski inequality for the surface area.In Chapter 4, we investigate the Lp mixed volume, dual Lp mixed volume. By strengthening Holder inequality and variational method, we obtained stability of Lp-Minkowski inequality and dual Lp-Minkowski inequality.