【作者】 郭婷；

【导师】 李小燕；

【作者基本信息】 湖南师范大学 ， 基础数学， 2010， 硕士

【摘要】 本文主要以对偶Brunn-Minkowski理论为基础,星体为研究对象,运用凸体几何知识和泛函分析方法,结合数学领域中具有较高应用价值的Clarkson不等式、Bellman不等式、Minkowski积分不等式、Holder积分不等式等解析不等式,建立了一系列对偶均质积分不等式。对偶均质积分是对偶混合体积的一种特殊形式,与经典理论中的均质积分相对应。本文第一章介绍凸体几何的发展历史和基本概念,并给出主要结果；第二章建立了商星体的对偶均质积分不等式,补充了n维欧氏空间中星体的几个简单且对称的对偶均质积分不等式；第三章推广了对偶仿射均质积分的对偶Brunn-Minkowski不等式；第四章建立了对偶仿射均质积分关于混合相交体的对偶Minkowski不等式和对偶Brunn-Minkowski不等式,以及给出了混合p-对偶仿射均质积分关于调和Blaschke线性组合的对偶Brunn-Minkowski不等式。更多还原

【Abstract】 This paper based on the dual Brunn-Minkowski theory, stars as the research object, used convex body geometry theory and functional analysis method, combined with Clarkson inequality、Bellman inequality、Minkowski integral inequality、Holder integral inequality such as analytic inequality, established a series of dual quermassintegrals inequalities.The dual quermassintegrals is a special form of mixed volume, corresponding to quermassintegrals of the classical theory. The development survey and main re-search directions of convex geometry are presented in the preface, and the main result were given. In Chapter 2, we set up the dual quermassintegrals inequalities of the quotient star bodies. In addition, a few simple and symmetric dual quermass-integrals inequalities in n-dimensional Euclidean space were completed. In Chapter 3, we presented the extensions of dual Minkowski inequality of the dual quermass-integrals. In Chapter 4, established the dual Minkowski inequality and the dual Brunn-Minkowski inequality of the dual quermassintegrals, which is relate to the mixed intersection body, and the dual Brunn-Minkowski inequality of the mixed p-dual affine quermassintegrals about reconciling Blaschke linear combination of were given.更多还原