对偶平均Minkowski非对称度临界点集的若干性质

发布时间：2018-03-13 02:19

本文选题：Minkowski对称度　切入点：对偶平均Minkowski对称度　出处：《苏州科技大学》2017年硕士论文　论文类型：学位论文

【摘要】：本论文主要研究了Q.Guo引进的一类新的凸体几何仿射不变量---对偶平均Minkowski对称度的临界点集。为揭示对偶平均Minkowski对称度和经典Minkowski对称度之间的关系,我们给出了Minkowski临界点处对偶平均Minkowski对称度的精确值。为得到此精确值,我们首先建立了有关半空间族的分析形式和几何形式的Helly型定理,在一定条件下,得到了一族半空间具有非空交的充分必要条件。然后,我们证明了,若一个凸体具有关于对偶平均Minkowski对称度的正则点,那么这个凸体存在过此临界点的n(10)1条仿射直径,从而在一定条件下肯定地回答了Grünbaum于1963年提出的一个猜想。本文得到的主要的成果如下:(1).分别以分析形式和几何形式给出了关于部分闭半空间的Helly型定理;(2).给出了Minkowski临界点处对偶平均Minkowski非对称度计算公式;(3).给出了凸体K有n(10)1个相交于一点的仿射直径的一个充分条件。
[Abstract]:In this paper, we mainly study the critical point set of a new class of geometric affine invariants of convex bodies introduced by Q. Guo-dual average Minkowski symmetry. In order to reveal the relationship between dual average Minkowski symmetry and classical Minkowski symmetry, In this paper, we give the exact value of dual average Minkowski symmetry at Minkowski critical point. In order to obtain the exact value, we first establish the Helly type theorem about the analytic form and geometric form of half-space family. A necessary and sufficient condition for a family of half-spaces to have a nonempty intersection is obtained. Then, we prove that if a convex body has a regular point for the dual average Minkowski symmetry degree, then the convex body has an affine diameter of n ~ (10) n ~ (10) over this critical point. Thus a conjecture put forward by Gr 眉 nbaum in 1963 is answered positively under certain conditions. The main results obtained in this paper are as follows: 1. The Helly type theorem on partially closed half-spaces is given in analytic form and geometric form respectively. A formula for calculating the dual average Minkowski asymmetry at the Minkowski critical point is presented. A sufficient condition for the affine diameters of a convex body K with n ~ (10)) intersecting at one point is given.
【学位授予单位】：苏州科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O186.5

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