关于R~n中凸体的Hadwiger包含问题和Bonnesen型不等式的几点注记
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摘要
回顾积分几何的发展历史,凸几何一直以来都是其研究的一个重要领域.凸体具有很多优美的性质,对它们的研究能够使我们发现和认识到其几何不变量之间的关系.概率和分析则是研究积分几何的重要工具.在研究欧氏空间R~n中一个凸体包含另一个凸体的充分条件方面,1941年德国数学家Hadwiger解决了平面时的情况,所以也被称为凸体的Hadwiger包含问题.但凸体的Hadwiger包含问题的高维情况的推广比较复杂,结果还在不断的完善.例如,周家足在《R~4中的Willmore泛函与包含问题》中给出了4维情况下凸体的Hadwiger包含问题的一个充分条件.又如,张高勇在他近期的论文《Geometric inequalities and inclusion measures of convexbodies》中给出了n维情况下凸体的Hadwiger包含问题的一个充分条件.等周问题一直以来也是积分几何研究的一个重要领域,并且它和包含问题有着密切的联系.本文首先给出了欧氏空间R~n中凸体的Hadwiger包含问题的另外一个充分条件及与其相关的Bonnesen型不等式;其次研究证实了常曲率曲面的一些Bonnesen型不等式;最后,给出了欧氏空间R~n中等周亏格与域的体积、面积之间的两个关系式.
本文的主要结论如下:
推论2.6.假设K和L是R~n中的凸体,K的面积和体积分别为A(K),V(K),L的体积和平均宽度分别为V(L),M(L).当V(K)≥V(L)时,经过一个等距变换使得L(?)K的充分条件是
当n为2维情况时,这个推论就是著名的凸集的Hadwiger包含定理.
定理2.10(Bonnesen-type不等式).如果rK和RK分别是R~n中凸体K的最大内接球半径和最小外接球半径,K的面积和体积分别为A(K),V(K).则有
定理3.1.若K是具有常高斯曲率K的欧氏球面上的紧致凸集，其中K的周长，面积，最大内接球半径和最小外接球半径分别为P_K，A_K，r_K，R_K.则有以下不等式：
定理3.7.若K是具有常高斯曲率-λ的双曲平面上的紧致凸集，其中K的周长，面积，最大内接球半径和最小外接球半径分别为P_K,A_K,r_K,R_K.则有以下不等式当R_K＝r_K时，不等式转化为此时等式成立当且仅当K是一个双曲圆盘．
定理4.1. R~n中的单连通域D，表面积为A体积为V. D~*是它的凸包，表面积为A~*体积为V~*.域D的等周亏格为△（D）=A~n-n~nω_nV~(n-1)，其中ω_n是n维单位球的体积．则等式成立当且仅当域D是一个标准球．
定理4.2. R~n中的单连通域D，表面积为A体积为V. D~*是它的凸包，表面积为A~*体积为V~*域D的等周亏格为△（D）=A~n-n~nω_nV~(n-1)，其中ω_n是n维单位球的体积．如果A≥A~*，则其中常数C满足C＜(?)，等式成立当且仅当域D是一个标准球．
Reviewing the history of the integral geometry, convex geometry has always been studied as an important area. Convex bodies have lots of elegant nature, the researches make us discover and understand the connection of the geometric invariants of the domains involved. Probability and analysis are useful tools to study integral geometry. The sufficient conditions that one domain can contain another domain were studied, in 1942, Hadwiger worked out the case of plane and called Hadwiger's containment problem for convex bodies. But the analogue of Hadwiger's containment problem for higher dimensions were very complex, the conclusion isn't very perfect and continues develope. For example, in《The Willmore functional and the containment problem in R~4》, Jiazu Zhou obtained a new analogue of Hadwiger's condition for convex bodies in R~4, in《Geometric Inequalities and Inclusion Measures of Convex Bodies》, recently Gaoyong Zhang acquired a new analogue of Hadwiger's condition for convex bodies in R~n. The isoperimetric problem is also an important area of integral geometry, and it has very close relation with containment problem. In this thesis, we discuss another analogue of Hadwiger's condition for convex bodies in R~n and Bonnesen-type inequalities firstly. Then we proof some Bonnesen-type inequalities for surfaces of constant curvature. We obtain the area deficit and volume deficit by the isoperimetric deficit at last.
We get the following theorems:
Corollary 2.6. Suppose K and L be convex bodies in R~n, let A(K) be the area of K, V(K) the volume of K, V(L) the volume of L, M(L) the mean width of L. If V(K)≥V(L), then the following condition is sufficient to guarantee that L(?)K, up to an isometry,
In the plane, this corollary is well-known Hadwiger's condition for convex sets.
Theorem 2.10. If r_K and R_K are the inradius and outradius of a convex body K in R~n respectively, let A(K) be the area of K, V(K) the volume of K. Then
Theorem 3.1. Suppose K∈k(S_k~2), let P_K be the perimeter of K, A_K the area of K, r_K the inradius of K, R_K the outradius of K. Then the following inequality holds:
Theorem 3.7. Suppose K∈k{H_λ~2), let P_K be the perimeter of K, A_K the area of K, r_K the inradius of K, R_K the outradius of K. Then the following inequality holds:with equality holds if and only if K is a hyperbolic disc.
Theorem 4.1. Suppose A and V are the area and volume of a simply connected region D in R~n respectively. And let D~* be its convex hull with area A~* and volume V~*.Let△(D)=A~n-n~nω_nV~n be the isoperimetric deficit of D, whereω_n is the volume of the n-dimensional unit ball. Then we havewith the equality holds if D is a standard sphere.
Theorem 4.2. Suppose A and V are the area and volume of a simply connected region in R~n respectively. And let D~* be its convex hull with area A~* and volume V~*. Let△(D)=A~n-nω_nV~n be the isoperimetric deficit of D, whereω_n is the volume of the n-dimensional unit ball. If A≥A~*, we havewith C is a constant and satisfy C<(?), the equality holds if D is a standard sphere.
本文的主要结论如下:
推论2.6.假设K和L是R~n中的凸体,K的面积和体积分别为A(K),V(K),L的体积和平均宽度分别为V(L),M(L).当V(K)≥V(L)时,经过一个等距变换使得L(?)K的充分条件是
当n为2维情况时,这个推论就是著名的凸集的Hadwiger包含定理.
定理2.10(Bonnesen-type不等式).如果rK和RK分别是R~n中凸体K的最大内接球半径和最小外接球半径,K的面积和体积分别为A(K),V(K).则有
定理3.1.若K是具有常高斯曲率K的欧氏球面上的紧致凸集，其中K的周长，面积，最大内接球半径和最小外接球半径分别为P_K，A_K，r_K，R_K.则有以下不等式：
定理3.7.若K是具有常高斯曲率-λ的双曲平面上的紧致凸集，其中K的周长，面积，最大内接球半径和最小外接球半径分别为P_K,A_K,r_K,R_K.则有以下不等式当R_K＝r_K时，不等式转化为此时等式成立当且仅当K是一个双曲圆盘．
定理4.1. R~n中的单连通域D，表面积为A体积为V. D~*是它的凸包，表面积为A~*体积为V~*.域D的等周亏格为△（D）=A~n-n~nω_nV~(n-1)，其中ω_n是n维单位球的体积．则等式成立当且仅当域D是一个标准球．
定理4.2. R~n中的单连通域D，表面积为A体积为V. D~*是它的凸包，表面积为A~*体积为V~*域D的等周亏格为△（D）=A~n-n~nω_nV~(n-1)，其中ω_n是n维单位球的体积．如果A≥A~*，则其中常数C满足C＜(?)，等式成立当且仅当域D是一个标准球．
Reviewing the history of the integral geometry, convex geometry has always been studied as an important area. Convex bodies have lots of elegant nature, the researches make us discover and understand the connection of the geometric invariants of the domains involved. Probability and analysis are useful tools to study integral geometry. The sufficient conditions that one domain can contain another domain were studied, in 1942, Hadwiger worked out the case of plane and called Hadwiger's containment problem for convex bodies. But the analogue of Hadwiger's containment problem for higher dimensions were very complex, the conclusion isn't very perfect and continues develope. For example, in《The Willmore functional and the containment problem in R~4》, Jiazu Zhou obtained a new analogue of Hadwiger's condition for convex bodies in R~4, in《Geometric Inequalities and Inclusion Measures of Convex Bodies》, recently Gaoyong Zhang acquired a new analogue of Hadwiger's condition for convex bodies in R~n. The isoperimetric problem is also an important area of integral geometry, and it has very close relation with containment problem. In this thesis, we discuss another analogue of Hadwiger's condition for convex bodies in R~n and Bonnesen-type inequalities firstly. Then we proof some Bonnesen-type inequalities for surfaces of constant curvature. We obtain the area deficit and volume deficit by the isoperimetric deficit at last.
We get the following theorems:
Corollary 2.6. Suppose K and L be convex bodies in R~n, let A(K) be the area of K, V(K) the volume of K, V(L) the volume of L, M(L) the mean width of L. If V(K)≥V(L), then the following condition is sufficient to guarantee that L(?)K, up to an isometry,
In the plane, this corollary is well-known Hadwiger's condition for convex sets.
Theorem 2.10. If r_K and R_K are the inradius and outradius of a convex body K in R~n respectively, let A(K) be the area of K, V(K) the volume of K. Then
Theorem 3.1. Suppose K∈k(S_k~2), let P_K be the perimeter of K, A_K the area of K, r_K the inradius of K, R_K the outradius of K. Then the following inequality holds:
Theorem 3.7. Suppose K∈k{H_λ~2), let P_K be the perimeter of K, A_K the area of K, r_K the inradius of K, R_K the outradius of K. Then the following inequality holds:with equality holds if and only if K is a hyperbolic disc.
Theorem 4.1. Suppose A and V are the area and volume of a simply connected region D in R~n respectively. And let D~* be its convex hull with area A~* and volume V~*.Let△(D)=A~n-n~nω_nV~n be the isoperimetric deficit of D, whereω_n is the volume of the n-dimensional unit ball. Then we havewith the equality holds if D is a standard sphere.
Theorem 4.2. Suppose A and V are the area and volume of a simply connected region in R~n respectively. And let D~* be its convex hull with area A~* and volume V~*. Let△(D)=A~n-nω_nV~n be the isoperimetric deficit of D, whereω_n is the volume of the n-dimensional unit ball. If A≥A~*, we havewith C is a constant and satisfy C<(?), the equality holds if D is a standard sphere.
引文
[1] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993.
[2] Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer, Berlin,1980.
[3] D. Ren, Topics in integral geometry, World Scientific Publishing Co., Inc., River Edge, NJ, 1994.
[4] L. A. Santal(?), Integral Geometry and Geometric Probability, Addison- Wesley, 1976.
[5] C. C. Hsiung, A First Coures in Differential Geometry, International Press, 1997.
[6] E. Grinberg, D. Ren, J. Zhou, The symmetric isoperimetric deficit and the containment problem in a plane of constant curvature, Preprint.
[7] J. Zhou, F. Chen, The Bonnesen-type inequalities in a plane of constant curvature, J. Korean Math. Soc., 2007, 44: 1363-1372.
[8] G. Zhang, Geometric inequalities and inclusion measures of convex bodies,Preprint.
[9] R. Osserman, Bonnesen-style isoperimetric inequality, Amer. Math. Monthly,1979, 86: 1-29.
[10] 任德麟,积分几何引论,上海科学出版社,上海,1988.
[11] R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc., 1978, 84: 1182-1238.
[12] G. Zhang, A sufficient condition for one convex body containing another, Chin. Ann. of Math., 1988, (4): 447-451.
[13] G. Zhang, Geometric inequalities and inclusion measures of convex bodies, Mathematika, 1994, 41: 95-116.
[14]J.Zhou,Kinematic formulas for mean curvature powers of hypersurfaces and Hadwiger's theorem in R~(2n),Trans.Amer.Math.Soc,1994,345:243-262.
[15]W.Y.Hsiang,An elementary proof of the isoperimetrie problem,Ann.of Math.,2002,23A(1):7-12.
[16]E.Grinberg,G.Zhang,J.Zhou,and S.Li,Integral Geometry and Convexity,World Sci.Publ.,2006.
[17]J.Zhou,The Willmore functional and the containment problem in R~4,Science in China Series A:Mathematics,2007,50(3):325-333.
[18]T.M.Apostol,Calculus,John Wiley and Sons,1969,2.
[19]J.Stillwell,Geometry of Surfaces,Springer-Verlag,New York,1992.
[20]J.Zhou,On Bonnesen-type inequalities,Act.Math.Sin.,2007,50(6):1397-1402.
[21]H.Federer,Geometric Measure Theory,Springer-Verlag,1969.
[22]H.Hadwiger,Uberdeckung ebener Bereiche durch Kreise und Quadrate,Comment.Math.Helv.,1941,13:195-200.
[23]H.Hadwiger,Gegenseitige Bedeckbarheit zweier Eibereiche und Isoperimetrie,Vierteljschr.Naturforsch.Gesellsch.Zurich,1941,86:152-156.
[24]D.Klain,G.-C.Rota,Introduction to Geometric Probability,Cambridge Univ.Press,New York,1997.
[25]D.Klain,Isometry invariant valuations on hyperbolic space,Discrete Comput.Geom.,2006,36(3):457-477.
[26]D.Klain,Bonnesen-type inequalities for surfaces of constant curvature,Adv.in Appl.Math.,2007,39:134-154.
[27]J.Zhou,On the plane isoperimetric deficit,Preprint.
[2] Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer, Berlin,1980.
[3] D. Ren, Topics in integral geometry, World Scientific Publishing Co., Inc., River Edge, NJ, 1994.
[4] L. A. Santal(?), Integral Geometry and Geometric Probability, Addison- Wesley, 1976.
[5] C. C. Hsiung, A First Coures in Differential Geometry, International Press, 1997.
[6] E. Grinberg, D. Ren, J. Zhou, The symmetric isoperimetric deficit and the containment problem in a plane of constant curvature, Preprint.
[7] J. Zhou, F. Chen, The Bonnesen-type inequalities in a plane of constant curvature, J. Korean Math. Soc., 2007, 44: 1363-1372.
[8] G. Zhang, Geometric inequalities and inclusion measures of convex bodies,Preprint.
[9] R. Osserman, Bonnesen-style isoperimetric inequality, Amer. Math. Monthly,1979, 86: 1-29.
[10] 任德麟,积分几何引论,上海科学出版社,上海,1988.
[11] R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc., 1978, 84: 1182-1238.
[12] G. Zhang, A sufficient condition for one convex body containing another, Chin. Ann. of Math., 1988, (4): 447-451.
[13] G. Zhang, Geometric inequalities and inclusion measures of convex bodies, Mathematika, 1994, 41: 95-116.
[14]J.Zhou,Kinematic formulas for mean curvature powers of hypersurfaces and Hadwiger's theorem in R~(2n),Trans.Amer.Math.Soc,1994,345:243-262.
[15]W.Y.Hsiang,An elementary proof of the isoperimetrie problem,Ann.of Math.,2002,23A(1):7-12.
[16]E.Grinberg,G.Zhang,J.Zhou,and S.Li,Integral Geometry and Convexity,World Sci.Publ.,2006.
[17]J.Zhou,The Willmore functional and the containment problem in R~4,Science in China Series A:Mathematics,2007,50(3):325-333.
[18]T.M.Apostol,Calculus,John Wiley and Sons,1969,2.
[19]J.Stillwell,Geometry of Surfaces,Springer-Verlag,New York,1992.
[20]J.Zhou,On Bonnesen-type inequalities,Act.Math.Sin.,2007,50(6):1397-1402.
[21]H.Federer,Geometric Measure Theory,Springer-Verlag,1969.
[22]H.Hadwiger,Uberdeckung ebener Bereiche durch Kreise und Quadrate,Comment.Math.Helv.,1941,13:195-200.
[23]H.Hadwiger,Gegenseitige Bedeckbarheit zweier Eibereiche und Isoperimetrie,Vierteljschr.Naturforsch.Gesellsch.Zurich,1941,86:152-156.
[24]D.Klain,G.-C.Rota,Introduction to Geometric Probability,Cambridge Univ.Press,New York,1997.
[25]D.Klain,Isometry invariant valuations on hyperbolic space,Discrete Comput.Geom.,2006,36(3):457-477.
[26]D.Klain,Bonnesen-type inequalities for surfaces of constant curvature,Adv.in Appl.Math.,2007,39:134-154.
[27]J.Zhou,On the plane isoperimetric deficit,Preprint.