摘要
均质积分被Minkowski提出,是凸体理论和积分几何中非常重要的概念和工具.Kubota、Cauchy、Steiner和很多的前辈对均质积分别给出了一系列的公式和定理.均质积分描述了凸体K和和它的正交投影K′_(n-r)之间的关系.凸体的外平行凸体是积分几何中的重要概念之一.在本文中,我们给出了关于凸体的正交投影的外平行凸体的几个性质.
这几个性质分别给出了(n-r)维空间L_(n-r[O])中的(n-r)维凸体K′_(n-r)的均质积分W′_i(K′_(n-r))和K′_(n-r)的外平行凸体(K′_(n-r))_ρ的表面积F((K′_(n-r))_ρ)、体积V((K′_(n-r))_ρ)、边界(?)((K′_(n-r))_ρ)的平均曲率积分M_i((?)((K′_(n-r))_ρ))与n维空间中的n维凸体K的均质积分之间关系.
本文的第一部分内容是关于几个三角恒等式的,这些恒等式是在证明和推广微分几何中经典的Euler公式时得到的.它们将应用于微分几何和积分几何中.但是作为三角恒等式,这些公式适合于更一般的情形.
定理1设三个角θ,,θ_i,θ_j满足:θ=±(θ_i-θ_j),或θ=θ_i+θ_j,或θ+θ_i+θ_j=2π,则有
注1此定理应用于Euler公式的证明中,是在证明Euler公式的过程中得到的一个初等公式.
定理2当θ=±(θ_i-θ_j)时,我们有当θ=θ_i+θ_j或θ+θ_i+θ_j=2π时,有
注2作为这些三角恒等式的特殊应用,Chen Zhao和Zhou给出了Euler公式和它关于的类推的简单证明(看参考文献[2]).
本文的第二部分是关于外平行凸体的Minkowski均质积分的,Kubota,Cauchy,Steiner等人给出了一系列的公式和定理.目前,武汉大学的李泽芳关于这部分作了很多工作,她的工作是把K′_(n-r)的外平行凸体限制在L_(n-r[O])平面上所作.下面是她最新的结论:
结论1设K为n维欧式空间E_n中的凸体,K_ρ是K的距离为ρ的外平行凸体,(K_ρ)′_(n-r)是K_ρ在(n-r)维平面L_(n-r[O])上所作的外平行凸体(即(K_ρ)′_(n-r)是K_ρ在(n-r)维平面L_(n-r[O])上的正交投影).W_(r+1+j)(K)是凸体K在n维欧式空间E_n中的均质积分,F((K_ρ)′_(n-r))表示(?)((K_ρ)′_(n-r))的表面积(即(n-r-1)维体积),则有
结论2设K为n维欧式空间E_n中的凸体,K_ρ是K的距离为ρ的外平行凸体,(K_ρ)′_(n-r)是K_ρ在(n-r)维平面L_(n-r[O])上所作的外平行凸体(即(K_ρ)′_(n-r)是K_ρ在(n-r)维平面L_(n-r[O])上的正交投影).W(r+i)(K)是凸体K在n维欧式空间E_n中的均质积分,V((K_ρ)′_(n-r))表示(K_ρ)′_(n-r)的(n-r)维体积,则有
结论3设K为n维欧式空间E_n中的凸体,K_ρ是K的距离为ρ的外平行凸体,(K_ρ)′_(n-r)是K_ρ在(n-r)维平面L_(n-r[O])上所作的外平行凸体(即(K_ρ)′_(n-r)是K_ρ在(n-r)维平面L_(n-r[O])上的正交投影).W_(r+s+1+j)(K)是凸体K在n维欧式空间E_n中的均质积分,M_s((?)(K_ρ)′_(n-r))表示(?)((K_ρ)′_(n-r))的第s个平均曲率积分,则有
本文的第二部分的主要工作分为两个方面:
一方面是把Kubota公式进行了推广,把它从(n-1)维空间L_(n-1[O])推广到(n-r)维空间L_(n-r[O])中;另一方面的工作是把李泽芳所作K′_(n-r)的外平行凸体限制在L_(n-r[O])平面上的这个限制去掉,而是在n维空间E_n中作的外平行凸体,从(n-r)维空间推广到了n维空间.
下面是我们在本文的第二部分外平行凸体的Minkowski均质积分中用到的定义和引理:
定义1(凸集、凸体、凸表面)
设K是n维欧式空间E_n中的一个子集,若当A,B∈K时,则连接二点的线段AB也属于K,就称K为E_n中的凸集.
紧致具有非空的内部的凸集称为凸体.
凸体K的边界(?)K称为凸表面.
注3今后我们仅限于讨论有界凸体.
注4以O_n表示n维单位球面的面积,可以表示为:
引理1 n维欧式空间E_n中过一个定点的非定向的r维平面的总测度(即:Grassmann流形G_(r,n-r)的体积)为其中O_i是i维单位球面的面积.
定义2(Minkowski均质积分)
设K为n维欧式空间E_n中的凸体,O是E_n中的一个定点.L_(n-r[O])表示过点O的任一(n-r)维平面.过K的每点作垂直于L_(n-r[O])的r维平面,这些r维平面与L_(n-r[O])的交点构成凸体K′_(n-r).K′_(n-r)叫做K到L_(n-r[O])上的正交投影,K′_(n-r)的(n-r)维体积记为V(K′_(n-r)).因为过定点的所有(n-r)维平面L_(n-r[O])形成一个Grassmann流形G_(n-r,r)我们引入下面的积分此式对r=1,…,n-1给出了I_r(K)定义.另外,补充规定:I_0(K)=V(K)(K的n维体积).(7)利用Grassmann流形G_(n-r,r)的体积公式(5),我们得到投影K′_(n-r)的体积V(K′_(n-r))的积分平均值:
E_n中凸体K的Minkowski均质积分W_r(K)定义如下:
1.当r=1,…,n-1时,或
2.当r=0时,
3.当r=n时,
定义3(外平行凸体、平行曲面)
设K为n维欧式空间E_n中的凸体,以K中每一点为球心、以常数ρ为半径作闭球体,这些球体的并集称为K的距离为ρ的外平行凸体,记为K_ρ.
K_ρ的边界(?)K_ρ称为边界(?)K的距离为ρ的平行曲面.
定义4(平均曲率积分)
设∑是n维欧式空间E_n中的一个C~2类的超曲面,k_1,k_2,…,k_(n-1)分别是∑的(n-1)个主曲率(函数),∑的第r个平均曲率积分(记为M_r(∑))定义为:其中d_σ表示∑的面积元,{k_(i_1),k_(i_2),…,k_(i_l)}为主曲率的第r阶初等对称函数.另外,补充规定:M_0(∑)=F(即∑的面积).(14)
乘积k_1k_2…k_(n-1)称为曲面的Gauss-Kronecker曲率,它与曲面的球面象的面积元du_(n-1)的联系是:其中d_σ为曲面∑的面积元.
R_i=1/(k_i)(i=1,…,n-1)称为∑的主曲率半径.从而平均曲率积分可以用主曲率半径定义如下:其中{R_(i_1),R_(i_2),…,R_(i_(n-r-1))}为R_(i_1),R_(i_2),…,R_(i_(n-1))的第(n-1)-r阶初等对称函数.
引理2(Kubota公式)
设K为n维欧式空间E_n中的凸体,L_(n-1[O])表示过点O的任一(n-1)维平面,K′_(n-1)为K在L_(n-1[O])上的投影.W_r(K)为n维欧式空间E_n中的凸体K的均质积分,W′_(r-1)(K′_(n-1))是(n-1)维空间L_(n-1[O])中的凸体K′_(n-1)的均质积分.则有其中(1/2)U_(n-1)表示(n-1)维单位球面的一半,即G_(1,n-1);Q_(n-2)表示(n-2)维单位球面的面积.
引理3(Cauchy公式)
设K为n维欧式空间E_n中的凸体,(?)K为K的凸表面,W_1(K)为K的均质积分,F为(?)K的表面积(即(n-1)维体积),则有F=nW_1(K).(16)
引理4(Steiner公式)
设K为n维欧式空间E_n中的凸体,K_ρ为K的距离为ρ的外平行凸体,V(K_ρ)表示K_ρ的体积,W_j(K)表示K的均质积分,则有
引理5设K为n维欧式空间E_n中的凸体,K_ρ为K的距离为ρ的外平行凸体,W_(i+j)(K)为K的均质积分,W_i(K_ρ)为K_ρ的均质积分,则有
引理6设n维欧式空间E_n中的凸体K的边界(?)K是一个C~2类的超曲面,K_ρ为K的距离为ρ的外平行凸体,(?)K为(?)K的距离为ρ的平行曲面.W_(r+1)(K)为K的均质积分,M_r((?)K)为(?)K的平均曲率积分,M_i((?)K_ρ)为(?)K_ρ的平均曲率积分,则有
运用上面的引理,我们可以得到如下定理:
定理3设K为n维欧式空间E_n中的凸体,K′_(n-r)为K到(n-r)维平面L_(n-r[O])上的正交投影.W_(i+r)(K)表示K在n维欧式空间E_n中的均质积分,W′_i(K′_(n-r))为(n-r)维空间L_(n-r[O])中的凸体K′_(n-r)的均质积分,则有其中O_i是i维单位球面的面积,G_(r,n-r)表示Grassmann流形.
注5当r=1时,定理3就是Kubota公式.
定理4设K为n维欧式空间E_n中的凸体,K′_(n-r)为K到(n-r)维平面L_(n-r[O])上的正交投影,(K′_(n-r))_ρ为K′_(n-r)在n维欧式空间E_n中的距离为ρ的外平行凸体.W_(j+1+r)(K)表示凸体K在n维欧式空间E_n中的均质积分,F((K′_(n-r))_ρ)为(?)((K′_(n-r))_ρ)的表面积(即(n-1)维体积),则有其中O_i是i维单位球面的面积,G_(r,n-r)表示Grassmann流形.
定理5设K为n维欧式空间E_n中的凸体,K′_(n-r)为K到(n-r)维平面L_(n-r[O])上的正交投影,(K′_(n-r))_ρ为K′_(n-r)在n维欧式空间E_n中的距离为ρ的外平行凸体.W_(j+r)(K)表示凸体K在n维欧式空间E_n中的均质积分,V((K′_(n-r))_ρ)表示(K′_(n-r))_ρ的n维体积,则有其中O_i是i维单位球面的面积,G_(r,n-r)表示Grassmann流形.
定理6设K为n维欧式空间E_n中的凸体,K′_(n-r)为K到(n-r)维平面L_(n-r[O])上的正交投影,(K′_(n-r))_ρ为K′_(n-r)在n维欧式空间E_n中的距离为ρ的外平行凸体,(?)((K′_(n-r))_ρ)为(?)(K′_(n-r))的距离为ρ的平行曲面,且(?)((K′_(n-r))_ρ)为C~2类的超曲面.W_(i+j+1+r)(K)表示凸体K在n维欧式空间E_n中的均质积分,M_i((?)((K′_(n-r))_ρ))表示(?)((k′_(n-r))_ρ)的平均曲率积分,则有其中O_i是i维单位球面的面积,G_(r,n-r)表示Grassmann流形.
The concept of quermassintegrale is introduced by Minkowski.It is of basic importance in the theory of convex bodies and integral geometry.Kubato,Cauchy, Steiner and others give a serial of formulas and theorems about the quermassinte-grale.Quermassintegrale describe the relationships between the convex figure K and its orthogonal projection K'_(n-r).In this Paper,we give several properties about the outer parallel convex body of the convex-figure's orthogonal projection.The properties respectively give the relations among quermassintegrale W_(i+r)(K)of the convex figure K,quermassintegrale W'_i(K'_(n-r))of the orthogonal projection K'_(n-r)of K and the(n-1)-dimensional surface area F((K'_(n-r))_ρ)and the n-dimensional volume V((K'_(n-r))_ρ)of the outer parallel body(K'_(n-r))_ρof the orthogonal projection K'_(n-r)as a convex figure in(n-r)-dimensional space L_(n-r[O])and integral of mean curvature M_i((?)((K'_(n-r))_ρ)).
In the first part of the thesis,we give several triangle identical equations.These triangle identical equations describe the relationships amongθ_k(k=i,j)that are the angles between the curvature vector kβof the intersection curveΓof two surfaces∑_k(k=i,j),the two surfaces∑_k(k=i,j)and the angleθbetween∑_i and∑_j.As the triangle identical equations,these formulas are fit for the general conditions.
Let the three anglesθ,,θ_i,θ_j satisfy the following equations:θ=±(θ_i-θ_j),orθ=θ_i+θ_j,orθ+θ_i+θ_j=2π,then
Whenθ=±(θ_i-θ_j),we have
Whenθ=θ_i+θ_j,orθ+θ_i+θ_j=2π,we get
In the second part of the thesis,we give several properties about the Minkowski quermassintegrale of the outer parallel body of the convex-figure's orthogonal pro-jection.
Let K be a convex figure in E_n and let O be a fixed point.Let L_(n-r[O])be all the(n-r)-planes through O and let K'_(n-r)be the orthogonal projection of K into the L_(n-r[O]).If W_(i+r)(K)are the quermassintegrale of a convex figure K in E_n and W'_i(K'_(n-r))are the quermassintegrale of K'_(n-r)as a convex figure in(n-r)-dimensional space L_(n-r[O]),then
Let K be a convex figure in E_n and let O be a fixed point.Let L_(n-r[O])be all the(n-r)-planes through O and let K'_(n-r)be its orthogonal projection of K into the L_(n-r[O]).Denoted by(K'_(n-r))_ρthe outer parallel body of K'_(n-r)in the distance p in E_n.If W_(j+1+r)(K)are the quermassintegrale of a convex figure K in E_n and F((K'_(n-r))_ρ)denotes the(n-1)-dimensional surface area of(?)((K'_(n-r))_ρ),then
Let K be a convex figure in E_n and let O be a fixed point.Let L_(n-r[O])be all the (n-r)-planes through O and let K'_(n-r)be its orthogonal projection of K into the L_(n-r[O]).Denoted by(K'_(n-r))_ρthe outer parallel body of the orthogonal projection K'_(n-r)in the distanceρin E_n.If W_(j+r)(K)are the quermassintegrale of a convex figure K in E_n and V((K'_(n-r))_ρ)denotes the n-dimensional volume of(K'_(n-r))_ρ,then
Let K be a convex figure in E_n and let O be a fixed point.Let L_(n-r[O])be all the (n-r)-planes through O and let K'_(n-r)be its orthogonal projection of K into the L_(n-r[O])and let(?)((K'_(n-r))_ρ)be a hypersurface of class C~2 in E_n.Denoted by(K'_(n-r))_ρthe outer parallel body of the orthogonal projection K'_(n-r)in the distance p in E_n. If W_(i+j+1+r)(K)are the quermassintegrale of K in E_n and M_i((?)((K'_(n-r))_ρ))denotes the integral of mean curvatures of(?)((K'_(n-r))_ρ),then
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