本文選題：Minkowski對稱度　切入點(diǎn)：對偶平均Minkowski對稱度　出處：《蘇州科技大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：本論文主要研究了Q.Guo引進(jìn)的一類新的凸體幾何仿射不變量---對偶平均Minkowski對稱度的臨界點(diǎn)集。為揭示對偶平均Minkowski對稱度和經(jīng)典Minkowski對稱度之間的關(guān)系,我們給出了Minkowski臨界點(diǎn)處對偶平均Minkowski對稱度的精確值。為得到此精確值,我們首先建立了有關(guān)半空間族的分析形式和幾何形式的Helly型定理,在一定條件下,得到了一族半空間具有非空交的充分必要條件。然后,我們證明了,若一個(gè)凸體具有關(guān)于對偶平均Minkowski對稱度的正則點(diǎn),那么這個(gè)凸體存在過此臨界點(diǎn)的n(10)1條仿射直徑,從而在一定條件下肯定地回答了Grünbaum于1963年提出的一個(gè)猜想。本文得到的主要的成果如下:(1).分別以分析形式和幾何形式給出了關(guān)于部分閉半空間的Helly型定理;(2).給出了Minkowski臨界點(diǎn)處對偶平均Minkowski非對稱度計(jì)算公式;(3).給出了凸體K有n(10)1個(gè)相交于一點(diǎn)的仿射直徑的一個(gè)充分條件。
[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.
【學(xué)位授予單位】：蘇州科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O186.5

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本文編號：1604373