摘要
在本文中,我们主要讨论了三个问题.首先,d维欧氏空间R~d上凸体的双弦幂积分;其次,由平坦凸体的平均曲率积分的性质讨论了投影体的外平形体的平均曲率积分;最后,拓展了陈省身的一个积分公式.
本文第二章对凸体的双弦幂积分进行了研究.双弦幂积分的概念是在弦幂积分论基础上建立的一个新的概念,从某个角度看,弦幂积分是双弦幂积分的特殊情形.双弦幂积分所获得的几何信息更丰富.本文得到了双弦幂积分的一些重要不等式,得到了以下结果:
定理2.5设K是R_d上的凸体,m,n是非负整数,则
定理2.6设K是R_d上的凸体,m,n,p是非负整数且0≤m≤n≤p,则
定理2.7双弦幂积分有下列不等式成立:
特别地,当m>n.
定理2.8双弦幂积分有下列不等式成立:
定理2.9双弦幂积分有下列不等式:当n是整数时,特别地,当n是偶数时,当n是奇数时,
在本文的第三章,我们讨论了R~d上凸体在L_r上正交投影后然后再做外平行体的平均曲率积分问题.这也是一类很有意思的问题,Santal(?)、周家足教授、江德烁、李泽芳等都进行过研究.其中,周家足教授和江德烁研究了在R~d中凸体先做外平行体再往平面L_r上作投影体的平均曲率积分问题,作者受到他们的启示研究了在R~d中凸体先往平面L_r上作投影体再在平面L_r上做外平行体的平均曲率积分问题,这是两个不同的问题,得到的结果也不一样,作者得到了下列定理:
定理3.4设K为d维欧氏空间E~d中具有C~2光滑边界(?)K的凸体,K'_r为r维平面L_r(?)E~d的投影体,(K'_r)_ρ为K'_r在E~d中的外平行体.M_i~(r)((?)(K'_r))(i=0,1…,r-1)是K'_r作为平坦凸体的平均曲率积分,令M_i~(d)((?)(K'_r)_p,)(i=0,1…,d-1)是(?)(K'_r)_ρ。在E~d中的平均曲率积分且(?)(K~r)_ρ∈C~2.因此,我们有
1)当id-r-1,则
其次,设△是两过固定o点的相交线性子空间所夹的夹角,△在相交线性子空间上的积分扮演着非常重要的角色,这属于积分几何中一类重要的问题:用已知的几何不变量来清楚地表示几何量关于运动的密度的积分.在本文的第四章,我们拓展了陈省身公式(为上述积分中的一个),得到了两相交线性子空间夹角的任意次幂在相交线性子空间上的积分,即下列定理:
定理4.1设L_(q[0])是过定点O的固定的q维平面,L_(q[0])是过O点活动的p维平面.设p+q-d>0,△是这两个线性子空间的角度,dL_(d-q[0])~(2d-q-p)表示dL_(2d-q-p[0])的子空间dL_(d-p)。的密度,则我们有
In this paper, we mainly investigate three problems, one is that the double chord-power integrals of a convex body in R~d. Secondly, by using characters of the flattened convex body's mean curvature integrals, we discuss about the mean curvature integrals of the outer parallel body of a projected convex set in E~d. Finally, we extend a formula of S. S. Chen.
In the second chapter, we study the double chord-power integrals of a convex body in R~d. The concept of double chord-power integrals is a new concept on the base of chord-power integrals. The chord-power integrals is a special case of the double chord-power integrals. And double chord-power integrals get more geometric information. In this paper we obtain the following geometric inequalities:
Theorem 2.5. Let K be a convex body in R~d, m, n are non-negative integer, then
Theorem 2.6. Let K be a convex body in R~d, m, n,p are non-negative integer and 0≤m≤n≤p, then
Theorem 2.7. The double chord-power integrals have the inequalitiesSpecial casewhen m > n,
Theorem 2.8. The double chord-power integrals have the inequality
Theorem 2.9. The double chord-power integrals have the inequalities:when n is integer.especially, when n is odd.when n is even.
In the third chapter we discuss the mean curvature integrals of a projected convex set of the outer parallel body of in E~d. This is a interesting problem, Santal(?) ,Professor Zhou, Jiang Deshou, Li Zefang and so on investigate the problem, especially, Professor Zhou and Jiang Deshou study the mean curvature integrals of the outer parallel body of a projected convex set in E~d. Author study the mean curvatureintegrals of a projected convex set of the outer parallel body of in E~d. This are two different problems, the results gotten is different, we obtain the following theorem:
Theorem 3.4. Let K be a convex body in E~d with C~2-smooth boundary (?)K,K~r be projection on the r-plane L_r (?) E~d, and (K~r)_ρbe the outer parallel body of K~T in the distanceρin E_d. M_i~(r)((?)(K~r))(i = 0, l,…,r-1) be the mean curvature integrals of (?)(K~R) as a convex surface of K~R and let M_I~(d)((?)(K~r)_ρ, )(i = 0,1,..., d-1) be the mean curvature integrals of (?)(K~r)_p as a flattened convex body of E~d and (?)(K~r)_p∈C~2. Then we have
1) If i < d - r - 1, then
where V_r(K~r) denotes the r-dimensional volume of K~r.
2) If i = d - r - 1, then
3) If i> d-r-1, then
Otherwise, Let△be the angle between two intersected linear subspaces through a fixed point O, the integral of the angle△over the intersected subspace play an important role in integral geometry, this integral is basic problem in integral geometry: find explicit formulas of the integrals of geometric quantities over the kinematic density in terms of known integral invariants. In the fourth chapter, We extend an integral formula of S. S. Chen (the problem is belong to the above integra)and obtain an integral of n-power of the angle of two intersected linear subspaces. Following, we introduce the theorem:
Theorem 4.1. Let L_(q[0]) be a fixed q-plane through a fixed point O and let L_(p[0]) be a moving p-plane through O.Assume that p + q > d. Let△be the angle between the two linear subspaces, express dL_(d-p[0])~(2d-p-q) be the density of dL_(d-p[0]) as a subspace of the fixed dL_(2d-p-q[0]), then we havewhere N is integer, O_i is the surface area of the i-dimension unit sphere.
引文
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