本文選題：平移包含測(cè)度　切入點(diǎn)：Minkowski不等式　出處：《西南大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：等周問題是幾何與凸幾何分析中的最經(jīng)典最重要的問題.等周不等式是幾何與分析中最重要的不等式之一.等周不等式與分析的Sobolev不等式等價(jià).Bonnesen型不等式是等周不等式的推廣和加強(qiáng).平面Bonnesen型不等式最近已經(jīng)被推廣到2維常曲率平面上.高維Bonnesen型不等式的研究一直是積分幾何與幾何不等式的困難問題,最近已有進(jìn)展.本文,將研究歐氏平面R~2中等周不等式以及Bonnesen型不等式的另一推廣,即關(guān)于平面兩凸域的Minkowski不等式以及Bonnesen型(Minkowski)對(duì)稱混合等似不等式.將估計(jì)歐氏平面R~2中一個(gè)凸域包含另一凸域的位似域的平移包含測(cè)度,估計(jì)凸域K0與K1的對(duì)稱混合等似虧格?2(K0,K1)=A201-A0A1(其中A0,A1分別是R~2中凸域K0,K1的面積,A01是K0與K1的混合面積).獲得了R~2中一個(gè)凸域包含另一凸域的位似域的充分條件,還得到了一些Bonnesen型對(duì)稱混合等似不等式和逆Bonnesen型對(duì)稱混合等似不等式,位似Bol-Fujiwara定理.我們還將研究n維歐氏空間Rn中由凸體K1,...,Kn所構(gòu)造的L_p混合質(zhì)心體,得到了關(guān)于L_p混合質(zhì)心體的一些幾何不等式.本文得到的這些結(jié)果是最新的.第3章主要研究平移包含測(cè)度.利用積分幾何中的運(yùn)動(dòng)公式,即Poincar′e平移運(yùn)動(dòng)公式和Blaschke平移運(yùn)動(dòng)基本公式,研究歐氏平面R~2中一凸域包含另一凸域的位似域的包含測(cè)度.得到了位似包含測(cè)度定理和平移包含測(cè)度定理.第4章主要研究歐氏平面R~2中兩凸域的對(duì)稱混合等似虧格?2(K0,K1)=A201-A0A1的上、下界.首先,定義一凸域關(guān)于另一凸域的內(nèi)半徑和外半徑,利用平移包含測(cè)度定理,得到一些Bonnesen型對(duì)稱混合等似不等式.特殊情況是:當(dāng)其中一個(gè)域?yàn)閳A盤時(shí),這些不等式就是歐氏平面R~2中周知的Bonnesen型等周不等式.我們還定義了一卵形域關(guān)于另一卵形域的曲率內(nèi)半徑和曲率外半徑,利用平移包含測(cè)度定理,得到了一些逆Bonnesen型對(duì)稱混合等似不等式.當(dāng)其中一個(gè)域?yàn)閳A盤時(shí),這些不等式就是歐氏平面R~2中的逆Bonnesen型等周不等式.本文中所獲得到的對(duì)稱混合等似不等式是歐氏平面R~2中關(guān)于兩凸域混合面積的Minkowski不等式的加強(qiáng).我們還得到了位似Bol-Fujiwara定理.第5章主要研究L_p混合質(zhì)心體.對(duì)n維歐氏空間Rn中以原點(diǎn)為內(nèi)點(diǎn)的n個(gè)凸體K1,...,Kn,我們定義了L_p混合質(zhì)心體Γp(K1,...,Kn),并得到關(guān)于L_p混合質(zhì)心體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.
【學(xué)位授予單位】：西南大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O186.5

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