导　　师： 周家足

学科专业： G0101

授予学位： 硕士

作　　者： ;

机构地区： 西南大学

摘　　要： 关于平面闭曲线的一个著名的问题是古典的等周问题：平面上面积固定的区域中,圆盘的周长最小.即平面上固定周长的简单闭曲线中,圆所围成的面积最大.也即 /(等周不等式/)平面简单闭曲线r所围成的区域的面积A,周长L满足：当且仅当r为圆时等号成立. 通常等周不等式都不涉及曲线的曲率,Gage给出了一个涉及平面凸曲线曲率平方积分的不等式.即： 若平面闭凸曲线的相对曲率为k,周长L,所围面积为A,则有当且仅当Γ为圆时等号成立. 关于平面上的Ros不等式是： 设平面卵形线r的曲率为k,周长与其所围成的面积分别为L,A则：当且仅当r为圆时等号成立. 本文第一部分主要研究平面上的卵形曲线的曲率积分不等式.首先,推导出一个函数的积分不等式并据此函数积分不等式得到了平面上Ros不等式的加强形式; 定理3.2.1设f为以2π为周期且具有二阶连续可微的函数,若f02πfdt=0,则等号成立当且仅当存在常数a,b使得f/(t/)=a cost+bsin t. 定理3.2.2设平面卵形线Γ的曲率为k,周长与其所围成的面积分别为L,A满足不等式：当且仅当Γ为圆时等号成立. 然后,用凸集外平行集的性质与函数的单调性给出了Gage等周不等式与曲率的熵不等式一种简化证明,并由此得到了一个新的有关曲率积分的等周不等式； 定理3.4.3设平面卵形线Γ的周长为L,曲率为k,则有当且仅当Γ为圆时等号成立. 再次,由Jensen不等式与卵形线的支持函数得到了一系列弱的曲率积分不等式； 定理3.5.2设平面卵形线Γ的周长为L,曲率为k,f/(x/)为定义在/(0,+∞/)上的严格凸函数,则有当且仅当Γ为圆时等号成立. 最后,本文还对卵形线的曲率序列积分做了进一步的讨论. 定理3.6.1设平面上卵形线r的周长为L,曲率为则以上每一个等号成立的充要条件是Γ为圆. 定理3.6.2设平面卵形线Γ的周长为L,曲率为则以上每一个等号成立的充要条件是Γ为圆. 定理3.6.3设平面卵形线Γ的周长为L,曲率为则对于任意满足1≤l≤m≤n的整数l,m,n都有以上每一个等号成立的充要条件是r为圆. 事实上,比经典的等周问题更基础的一个问题是关于多边形的等周问题.对于平面多边形,也有相应的离散型的等周不等式： /(多边形的等周不等式/)设Γn为欧氏平面R2中面积为An,周长为Ln的n边形,则有不等式：等号成立的充分必要条件是r。为正n边形. 一般地,△/(Γ/)=L2-4πA称r所围区域的等周亏格.等周亏格用来度量周长为L,面积为A的区域与半径为L//2π圆的差别程度.类似地,我们可以定义多边形Γn的等周亏格为△/(Γn/)=L2-4cnA,用来度量Γn与正n边形的差别程度. 本文的第二部分主要讨论平面多边形的等周亏格上下界的估计.首先,根据X.M.Zhang的方法得到了两个平面多边形的Bonnesen型不等式： 定理4.2.5设Γn为欧氏平面R2中面积为An,周长为Ln,内接于半径为R的圆的n边形,是半径为R的圆的内接正多边形的面积,则有不等式：其中,等号成立的充分必要条件是Γn为正n边形. 再次,由平面多边形几何量之间的关系得到了一组平面多边形的Bottema型不等式： 定理4.3.4设r。是面积为An,周长为L。的凸n边形rire分别为Γn的最大内接圆半径与最小外接圆半径,则有如下不等式成立.以上各式等号成立的充分必要条件Γn为正n边形. As a well known result, The isoperimetric theorem states that:for a simple closed curveΓ/(in the euclidian plane/) of length L enclosing a domain of area A, we have the inequality Equality is attained if and only if this curve is a euclidean circle. This means that among the set of domains of fixed area, the euclidean circle has the smallest perimeter or among the set of domains of fixed perimeter, the euclidean circle has the biggest area. In general, The isoperimetric inequality do not involve the curvature, Then, Gage gives integral inequality involving the square of the curvature for convex curve. That LetΓbe a oval curve of length L in the plane R2, and A the area enclosed byΓ. Then the curvatureκofΓsatisfy the following inequality: Equality holds if and only ifΓis a circle. The Ros isoperimetric theorem states in the plane that: LetΓbe a oval curve in the plane R2, and A the area enclosed byΓ. Then the curvatureκofΓsatisfy the following inequality: Equality holds if and only ifΓis a circle. In the first part of this paper, we investigate some flat oval curve of curvature integral inequalities. First, an integral inequality of function is derived, and on the basis of this integral inequality strengthen Ros Inequality form has been obtained; Theorem 3.2.1 Let f be is a C2-function of period 2πand f02πfdt=0, then Equality holds if and only if f/(t/)=a cos t+b sin t,where a, b are constants. Theorem 3.2.2 LetΓbe a oval curve of length L in R2, and A the area enclosed byΓ. Then the curvatureκofΓsatisfy the following inequality: Equality holds if and only ifΓis a circle. Next, by the properties of outside parallel convex set and monotonicity of the function a simple proof of Gage Isoperimetric Inequalities and Entropy Inequality for curvature are given, and thus we obtain a new integral on the curvature of the isoperimetric inequality; Theorem 3.4.3 LetΓbe a oval curve of length L in the plane R2, and A the area enclosed byΓ. Then the curvatureκofΓsatisfy the following inequality: Equality holds if and only ifΓis a circle. By Jensen inequality and the oval support functions we obtain a series of weak curvature integral inequalities; Theorem 3.5.2 LetΓbe a oval curve of length L in the plane R2, and f/(x/) is a strictly convex function on /(0,+∞/). Then the curvatureκofΓsatisfy the following inequality: Equality holds if and only ifΓis a circle. Continuously, we do some discussion about series of the curvature integral inequalities for oval in the paper. Theorem 3.6.1 LetΓbe a oval curve of length L in the plane R2. Then the curvatureκofΓsatisfy the following inequalities: Where Each equality sign holds if and only ifΓis a circle. Theorem 3.6.2 LetΓbe a oval curve of length L in the plane R2. Then the curvatureκof F satisfy the following inequalities: Where Each equality sign holds if and only ifΓis a circle. Theorem 3.6.3 LetΓbe a oval curve of length L in the plane R2, then the curvatureκofΓfor every positive integer 1≤l≤m≤n satisfy the following inequalities: Where Each equality sign holds if and only ifΓis a circle. In fact, There is a more basic problem for polygons than the classical isoperimetric problem.for polygons, There is also the corresponding discrete isoperimetric Inequality: /(the isoperimetric inequality for polygons/) LetΓn be an n-gon /(a polygon with n sides/) of perimeter Ln and area An, the following inequality is known Equality is attained if and only if the n-gon is regular. In geometry,△/(Γ/)= L2-4πA is called the isoperimetric deficit of the curveΓ. It measures the 'deviation ofΓfrom circularity'. Similarly, We can also define the isoperimetric deficit△/(Γn/)= L2-4cnA of the polygonΓn. It measures the 'deviation ofΓn from the n-gon is regular'. In the second part, we have a discussion about isoperimetric deficit for plane polygon. First of all, according to X. M. Zhang's idea two plane polygon's Bonnesen-type inequalities are obtained: Theorem 4.2.5 LetΓn be a convex polygon of length Ln in the plane R2, and An the area enclosed byΓn. Then the in-radius R ofΓn satisfy the following inequalities, there Each equality sign holds if and only ifΓn is a regular n-gon. Lastly, by geometrical relationship for the polygon some Bottema-type inequalities for planar polygons have been obtained: Theorem 4.3.4 LetΓn be a convex polygon of length Ln in R2, and An the area enclosed byΓn. Then in-radius ri and circum-radius re ofΓn satisfy the following inequalities, Each equality sign holds if and only ifΓn is a regular n-gon.

关 键 词： 等周不等式 等周不等式 不等式 卵形线 型不等式 型不等式 多边形

分 类 号： [TH1 O18]

领　　域： [机械工程] [理学] [理学]