Bonnesen型对称混合等似不等式与L_p混合质心体

发布时间：2018-03-16 22:33

本文选题：平移包含测度　切入点：Minkowski不等式　出处：《西南大学》2016年博士论文　论文类型：学位论文

【摘要】：等周问题是几何与凸几何分析中的最经典最重要的问题.等周不等式是几何与分析中最重要的不等式之一.等周不等式与分析的Sobolev不等式等价.Bonnesen型不等式是等周不等式的推广和加强.平面Bonnesen型不等式最近已经被推广到2维常曲率平面上.高维Bonnesen型不等式的研究一直是积分几何与几何不等式的困难问题,最近已有进展.本文,将研究欧氏平面R~2中等周不等式以及Bonnesen型不等式的另一推广,即关于平面两凸域的Minkowski不等式以及Bonnesen型(Minkowski)对称混合等似不等式.将估计欧氏平面R~2中一个凸域包含另一凸域的位似域的平移包含测度,估计凸域K0与K1的对称混合等似亏格?2(K0,K1)=A201-A0A1(其中A0,A1分别是R~2中凸域K0,K1的面积,A01是K0与K1的混合面积).获得了R~2中一个凸域包含另一凸域的位似域的充分条件,还得到了一些Bonnesen型对称混合等似不等式和逆Bonnesen型对称混合等似不等式,位似Bol-Fujiwara定理.我们还将研究n维欧氏空间Rn中由凸体K1,...,Kn所构造的L_p混合质心体,得到了关于L_p混合质心体的一些几何不等式.本文得到的这些结果是最新的.第3章主要研究平移包含测度.利用积分几何中的运动公式,即Poincar′e平移运动公式和Blaschke平移运动基本公式,研究欧氏平面R~2中一凸域包含另一凸域的位似域的包含测度.得到了位似包含测度定理和平移包含测度定理.第4章主要研究欧氏平面R~2中两凸域的对称混合等似亏格?2(K0,K1)=A201-A0A1的上、下界.首先,定义一凸域关于另一凸域的内半径和外半径,利用平移包含测度定理,得到一些Bonnesen型对称混合等似不等式.特殊情况是:当其中一个域为圆盘时,这些不等式就是欧氏平面R~2中周知的Bonnesen型等周不等式.我们还定义了一卵形域关于另一卵形域的曲率内半径和曲率外半径,利用平移包含测度定理,得到了一些逆Bonnesen型对称混合等似不等式.当其中一个域为圆盘时,这些不等式就是欧氏平面R~2中的逆Bonnesen型等周不等式.本文中所获得到的对称混合等似不等式是欧氏平面R~2中关于两凸域混合面积的Minkowski不等式的加强.我们还得到了位似Bol-Fujiwara定理.第5章主要研究L_p混合质心体.对n维欧氏空间Rn中以原点为内点的n个凸体K1,...,Kn,我们定义了L_p混合质心体Γp(K1,...,Kn),并得到关于L_p混合质心体Γp(K1,...,Kn)的一些重要不等式.
[Abstract]:Isoperimetric problem is the most classical and most important problem in geometric and convex geometric analysis. Isoperimetric inequality is one of the most important inequalities in geometry and analysis. Isoperimetric inequality is equivalent to the Sobolev inequality of analysis. Bonnesen-type inequality is isoperimetric. The extension and strengthening of inequality. The inequality of plane Bonnesen type has recently been extended to 2-dimensional plane of constant curvature. The study of high-dimensional Bonnesen type inequality has always been a difficult problem of integral geometry and geometric inequality. Recent advances have been made. In this paper, we will study another extension of the Euclidean plane Ry 2 Intermediate inequality and Bonnesen type inequality. In this paper, the Minkowski inequality for two convex domains in a plane and the symmetric mixing inequality for Bonnesen type Mimkowski2 are given. The translation inclusion measure of a convex domain containing another convex domain in the Euclidean plane R2 is estimated, and the symmetrically mixed isobaric genus of K0 and K1 is estimated. 2K0 / K1 / A201-A0A1 (where A0A1 is the area of the convex domain K0K1 in RK2 is the mixed area of K0 and K1. Sufficient conditions are obtained for one convex domain in R2 to contain a quasidomain of another convex domain. We also obtain some Bonnesen type symmetric mixed equivalent inequalities and inverse Bonnesen type symmetric mixed equivalent inequalities, and the potential Bol-Fujiwara theorem. We will also study the LSP mixed centroids constructed by convex K1C... Kn in n-dimensional Euclidean space R _ n, we will also study the LSP mixed centroids in n-dimensional Euclidean space. In this paper, we obtain some geometric inequalities for LP mixed centroids. These results are the latest. In Chapter 3, we mainly study the measure of translation inclusions, and use the kinematic formulas in integral geometry. That is, the Poincar'e translation motion formula and the Blaschke translation motion basic formula, In this paper, we study the inclusion measure of a convexity domain containing another convex domain in the Euclidean plane R2. We obtain a bit-like inclusion measure theorem and a translational inclusion measure theorem. In Chapter 4, we mainly study the symmetrically mixed isobaric genus of two convex domains in the Euclidean plane R ~ (2)? (2) the upper and lower bounds of K0 / K1 / A201-A0A1. Firstly, the inner and outer radii of a convex domain for another convex domain are defined. By using the translational inclusion measure theorem, some Bonnesen type symmetric mixing inequalities are obtained. The special case is: when one of the domains is a disk, These inequalities are known as Bonnesen type isoperimetric inequalities in the Euclidean plane R2. We also define the inner and outer radii of curvature for one ovate domain and the outer radius of curvature for another oval domain, and use the theorem of translational inclusion measure. In this paper, we obtain some inverse Bonnesen type symmetric mixed equality inequalities. When one of the domains is a disk, These inequalities are inverse Bonnesen's type isoperimetric inequalities in the Euclidean plane R2. The symmetric mixed isobaric inequalities obtained in this paper are the strengthening of Minkowski's inequality on the mixed area of two convex domains in the Euclidean plane R2. In chapter 5, we mainly study the mixed centroid of LP. For n convex bodies in n-dimensional Euclidean space R n with origin as the inner point, we define the mixed mass body 螕 pn K1n... We obtain some important inequalities about LP mixed centroid 螕 pPU K1n.
【学位授予单位】：西南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O186.5

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