一类混合相交体对偶均质积分的逆向Minkowski不等式

发布时间：2018-06-04 09:08

  本文选题：凸体 + 星体　； 参考：《西南大学》2017年硕士论文

【摘要】：经典的Brunn-Minkowski理论起源于H.Brunn的博士论文和H.Minkowski的开创性工作.经过 Bonnesen,Santalo,Fenchel,Blaschke,Busemann 等人的推动,使得凸几何理论日渐完善,成为一个单独的几何研究分支.而在1975年Lutwak发表的《Dual Mixed Volumes》一文中首次定义了星体,建立了与凸体相对应的对偶Brunn-Minkowski理论.对偶理论的建立,对解决Busemann-Petty问题有跨时代的意义.自20世纪80年代以来,凸体理论与星体理论的建立,使几何学家们对凸几何的研究取得重大成果,如 Lp-Brunn-Minkowski 和 Oricz-Brunn-Minkowski 理论.本文第一部分和第二部分介绍了研究背景和文中所需要的预备知识.本文第三部分和第四部分是文章的主旨内容,主要探究了混合相交体的Brunn-Minkowski理论中的不等式.在第三部分运用Polya-Szego不等式和Holder不等式证明了混合相交体均质积分的逆向Minkowski不等式,在此基础上将系数0≤i≤n,0≤j≤n-1扩充到0≤i,0≤j上.在第四部分运用逆向的Minkowski不等式证明了混合相交体对偶均质积分的逆向Brunn-minkowski不等式.
[Abstract]:The classical Brunn-Minkowski theory originates from H.Brunn 's doctoral thesis and H.Minkowski 's pioneering work. By Bonnesenn Santalofencheln Blaschkeke Busemann et al., the convex geometry theory is becoming more and more perfect and becomes a separate branch of geometry research. In 1975, Lutwak published "Dual Mixed Volumes" for the first time, defining stars and establishing the dual Brunn-Minkowski theory corresponding to convex bodies. The establishment of duality theory has a cross-epoch significance for solving Busemann-Petty problem. Since the eighties of the 20th century, the establishment of convex body theory and star theory has made great achievements in the research of convex geometry by geometrists, such as Lp-Brunn-Minkowski and Oricz-Brunn-Minkowski theory. The first and second parts of this paper introduce the research background and the preparatory knowledge needed in the paper. The third and fourth parts of this paper are the main content of the paper, mainly to explore the inequalities in the Brunn-Minkowski theory of mixed intersections. In the third part, Polya-Szego inequality and Holder inequality are used to prove the inverse Minkowski inequality for homogeneous integrals of mixed intersecting bodies. On this basis, the coefficient 0 鈮，

本文编号：1976808