关于凸体的Bonnesen型Aleksandrov-Fenchel不等式
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摘要
经典的等周不等式、Bonnesen型不等式和Aleksandrov-Fenchel不等式是几何学中非常重要的不等式.n维欧氏空间张域K的等周亏格△。(K)=An—nnωnVn-1(其中A为凸体K的面积,V为K的体积,ω。为n维单位球的体积)刻画了几何体K和球之间的差别程度Hadwige、Osserman、Klain、Bottema、 Zhou等都曾致力于用积分几何的方法研究等周亏格的上界和下界.本文主要研究两个问题.第一,研究常曲率平面x∈2中等周亏格的下界即Bonnesen型不等式.第二,利用著名的Aleksandrov-Fenchel不等式,研究欧氏空间Rn中两凸体K,L的对称混合位似亏格△i(K,L)和Aleksandrov-Fenchel亏格△i(K).其中,Aleksandrov-Fenchel等周亏格是等周亏格的推广,Bonnesen型Aleksandrov-Fenchel不等式即Aleksandrov-Fenchel亏格的下界是Bonnesen型不等式的推广.
常曲率平面X∈2中域K的等周亏格为△∈(K)=L2—4πA+∈A2(其中A为区域K的面积,L为K的周长,∈为常曲率平面的曲率,即∈=0时为欧氏平面R2；∈>0为射影平面PR2；∈<0为双曲平面H∈2),Klain、Zhou和Chen用不同的办法得到了X∈2中域K的等周亏格的下界.本文首先从一域包含另一域的包含测度思想出发,运用积分几何的基本运动公式,得到X∈2中域K的等周亏格的两个下界,这些下界加强了Zhou, Chen的结果.主要结果如下：
定理3.3.9设K为常曲率平面X∈2中面积为A,周长为L的严格凸区域,则其中,ri和r。分别为K的最大内接测地圆半径及最小外接测地圆半径,等号成立当且仅当K为测地圆盘.
定理3.3.11设K为常曲率平面X∈2中面积为A,周长为L的严格凸区域,则下列不等式成立：其中,ri和r。分别为K的最大内接测地圆半径及最小外接测地圆半径,等号成立当且仅当K为测地圆盘.
其中,tn∈;ct∈的定义参见(3-37)和(3-38).
著名的Aleksandrov-Fenchel不等式是Brunn-Minkowski理论中的重要内容.除经典的Aleksandrov-Fenchel不等式之外Lutwa、Yang、Zhang等还得到各种形式的Aleksandrov-Fenchel不等式.本文利用经典的Aleksandrov-Fenchel不等式,探讨Aleksandrov-Fenchel等周亏格△i(K)=Wi2(K)—Wi-1(K)Wi+1(K)(其中Wi(K)为凸体K的第i阶均质积分)和两凸体K,L的对称混合位似亏格△i(K,L)=Vi2(K,L)—Vi—1(K,L)V+1(K,L)(其中Vi(K,L)为K,L的第i阶混合体积),获得了Aleksandrov-Fenchel等周亏格的几个下界估计.作为它们的直接推论,得到了R2、R3中用投影体的宽度函数表示的等周亏格的下界.这有别于传统的用最大内接圆半径和最小外接圆半径表示的Bonnesen型不等式.本文最后根据混合体积的单调性,得到Aleksandrov-Fenchel等周亏格(等周亏格的推广)的上界.事实上,到目前为止关于等周亏格的上界估计的结果比较少.我们所得主要结果如下：
定理4.2.10设K为欧氏空间Rn中的凸体,Wi(K)为K的第i阶均质积分.则其中,u∈Sn-1,u⊥为与方向u垂直的线性子空间,Ku为凸体K向u上的正交投影,wn—1(Ku)为Ku的平均宽度,其最大和最小值分别记成Wmax(Ku)和Wmin(Ku).
定理4.3.2设K为欧氏空间Rn中的凸体,Wi(K)为K的第i阶均质积分.如果1≤i≤n-1,则△i(K)=Wi2(K)—Wi+1(K)Wi-1(K)≤Wi+1(K)Wi(K)(R(K,B)—r(K,B)),其中,r(K,B)和R(K,B)分别为K的相对最大内接圆半径和相对最小外接圆半径.
The classical isoperimetric inequality, the Bonnesen-style inequality and the Aleksandrov-Fenchel inequality are important inequalities in geometry. The isoperi-metric deficit measures the deficit between a geometric body and a disc. Had-wiger, Osserman, Klain, Bottema and Zhou and so on make efforts to investigate the lower bound and the upper bound of the isoperimetric deficit. In this paper, we mainly study two problems. Firstly, we investigate the Bonnesen-style inequal-ity of convex domain in a plane of constant curvature, and obtain analogues of the Bonnesen inequality that strengthen the known Bonnesen style inequalities. Secondly, using the known Aleksandrov-Fenchel inequality, we study the sym-metric mixed homothetic deficit△i(K, L) of convex bodies K and L in Rn and the Aleksandrov-Fenchel deficit△i(K). Moreover the Aleksandrov-Fenchel deficit is the generation of isoperimetric deficit, the Bonnesen Aleksandrov-Fenchel in-equality (the lower bound of Aleksandrov-Fenchel deficit) is the generation of Bonnesen inequality.
The isoperimetric deficit of K in a plane of constant curvature is△∈(K)=L2-4πA+∈A2(where K is the domain of area A and circum-length L,∈is the curvature of the plane of constant curvature), Klain, Zhou and Chen obtain sev-eral lower bounds of the isoperimetric deficit in a plane of constant curvature by differential methods. In this paper we first investigate the containment measure idea and the fundamental kinematic formula in integral geometry, and obtain two lower bounds of the isoperimetric deficit that strengthen the known isoperimetric deficit. We have the following theorems:
Theorem3.3.9Let K be a strictly convex domain of area A and circum-length L in X∈2, then where re, ri are the smallest geodesic circumradius and the biggest geodesic in-radius of K respectively. The equality sign holds if and only if K is a geodesic disc.
Theorem3.3.11Let K be a. strictly convex domain of area A and circum-length L in X∈2, then where re, ri are the smallest geodesic circumradius and the biggest geodesic in-radius of K respectively. The equality sign holds if and only if K is a geodesic disc.
The famous Aleksandrov-Fenchel inequality is very important in Brunn-Minkowski theory. Except the classical Aleksandrov-Fenchel inequality, Lutwak. Yang, Zhang obtain several different forms of Aleksandrov-Fenchel inequality. Us-ing the classical Aleksandrov-Fenchel inequality, we study the symmetric mixed homothetic deficit△i(K, L)(where△i(K, L)=Vi2(K, L)-Vi-1(K, L)Vi+1(K, L) is the mixed volum of K, L) and the Aleksandrov-Fenchel isoperimetric deficit△i(K)=Wi2(K)-Wi-1(K)Wi+1(K)(where Wi(K) i-th quermassintegral of K), and obtain several lower bounds of the isoperimetric deficit. As direct corollar-ies, we obtain the lower bounds of isoperimetric deficit expressed by the width function of project body in two-dimensional, three-dimensional Euclidean space. Finally, we obtain an upper bounds of isoperimetric deficit. The results are:
Theorem4.2.10Let K be a convex body in Rn and Wi(K) i-th quer-massintegral of K. Then where u∈Sn-1, u⊥is the linear subspace perpendicular to u, Ku is the orthogonal projection of a convex body K onto the linear subspace u⊥, ωd-1(Ku) is the mean width of Ku, and ωmax(Ku) and ωmin(Ku) are the maximum and minimum width of ω(Ku) over all u∈Sn-1.
Theorem4.3.2Let K be a convex body in Rn and Wi(K) i-th quermass-integral of K. If1≤i≤n-1, then△i(K)=Wi2(K)-Wi+1(K)Wi-1(K)≤Wi+1(K)Wi(K)(R(K, B)-r(K, B)), where r(K, B) and R(K, B) are the relative inradius and the relative circumradius of K with respect to B, respectively.
常曲率平面X∈2中域K的等周亏格为△∈(K)=L2—4πA+∈A2(其中A为区域K的面积,L为K的周长,∈为常曲率平面的曲率,即∈=0时为欧氏平面R2；∈>0为射影平面PR2；∈<0为双曲平面H∈2),Klain、Zhou和Chen用不同的办法得到了X∈2中域K的等周亏格的下界.本文首先从一域包含另一域的包含测度思想出发,运用积分几何的基本运动公式,得到X∈2中域K的等周亏格的两个下界,这些下界加强了Zhou, Chen的结果.主要结果如下：
定理3.3.9设K为常曲率平面X∈2中面积为A,周长为L的严格凸区域,则其中,ri和r。分别为K的最大内接测地圆半径及最小外接测地圆半径,等号成立当且仅当K为测地圆盘.
定理3.3.11设K为常曲率平面X∈2中面积为A,周长为L的严格凸区域,则下列不等式成立：其中,ri和r。分别为K的最大内接测地圆半径及最小外接测地圆半径,等号成立当且仅当K为测地圆盘.
其中,tn∈;ct∈的定义参见(3-37)和(3-38).
著名的Aleksandrov-Fenchel不等式是Brunn-Minkowski理论中的重要内容.除经典的Aleksandrov-Fenchel不等式之外Lutwa、Yang、Zhang等还得到各种形式的Aleksandrov-Fenchel不等式.本文利用经典的Aleksandrov-Fenchel不等式,探讨Aleksandrov-Fenchel等周亏格△i(K)=Wi2(K)—Wi-1(K)Wi+1(K)(其中Wi(K)为凸体K的第i阶均质积分)和两凸体K,L的对称混合位似亏格△i(K,L)=Vi2(K,L)—Vi—1(K,L)V+1(K,L)(其中Vi(K,L)为K,L的第i阶混合体积),获得了Aleksandrov-Fenchel等周亏格的几个下界估计.作为它们的直接推论,得到了R2、R3中用投影体的宽度函数表示的等周亏格的下界.这有别于传统的用最大内接圆半径和最小外接圆半径表示的Bonnesen型不等式.本文最后根据混合体积的单调性,得到Aleksandrov-Fenchel等周亏格(等周亏格的推广)的上界.事实上,到目前为止关于等周亏格的上界估计的结果比较少.我们所得主要结果如下：
定理4.2.10设K为欧氏空间Rn中的凸体,Wi(K)为K的第i阶均质积分.则其中,u∈Sn-1,u⊥为与方向u垂直的线性子空间,Ku为凸体K向u上的正交投影,wn—1(Ku)为Ku的平均宽度,其最大和最小值分别记成Wmax(Ku)和Wmin(Ku).
定理4.3.2设K为欧氏空间Rn中的凸体,Wi(K)为K的第i阶均质积分.如果1≤i≤n-1,则△i(K)=Wi2(K)—Wi+1(K)Wi-1(K)≤Wi+1(K)Wi(K)(R(K,B)—r(K,B)),其中,r(K,B)和R(K,B)分别为K的相对最大内接圆半径和相对最小外接圆半径.
The classical isoperimetric inequality, the Bonnesen-style inequality and the Aleksandrov-Fenchel inequality are important inequalities in geometry. The isoperi-metric deficit measures the deficit between a geometric body and a disc. Had-wiger, Osserman, Klain, Bottema and Zhou and so on make efforts to investigate the lower bound and the upper bound of the isoperimetric deficit. In this paper, we mainly study two problems. Firstly, we investigate the Bonnesen-style inequal-ity of convex domain in a plane of constant curvature, and obtain analogues of the Bonnesen inequality that strengthen the known Bonnesen style inequalities. Secondly, using the known Aleksandrov-Fenchel inequality, we study the sym-metric mixed homothetic deficit△i(K, L) of convex bodies K and L in Rn and the Aleksandrov-Fenchel deficit△i(K). Moreover the Aleksandrov-Fenchel deficit is the generation of isoperimetric deficit, the Bonnesen Aleksandrov-Fenchel in-equality (the lower bound of Aleksandrov-Fenchel deficit) is the generation of Bonnesen inequality.
The isoperimetric deficit of K in a plane of constant curvature is△∈(K)=L2-4πA+∈A2(where K is the domain of area A and circum-length L,∈is the curvature of the plane of constant curvature), Klain, Zhou and Chen obtain sev-eral lower bounds of the isoperimetric deficit in a plane of constant curvature by differential methods. In this paper we first investigate the containment measure idea and the fundamental kinematic formula in integral geometry, and obtain two lower bounds of the isoperimetric deficit that strengthen the known isoperimetric deficit. We have the following theorems:
Theorem3.3.9Let K be a strictly convex domain of area A and circum-length L in X∈2, then where re, ri are the smallest geodesic circumradius and the biggest geodesic in-radius of K respectively. The equality sign holds if and only if K is a geodesic disc.
Theorem3.3.11Let K be a. strictly convex domain of area A and circum-length L in X∈2, then where re, ri are the smallest geodesic circumradius and the biggest geodesic in-radius of K respectively. The equality sign holds if and only if K is a geodesic disc.
The famous Aleksandrov-Fenchel inequality is very important in Brunn-Minkowski theory. Except the classical Aleksandrov-Fenchel inequality, Lutwak. Yang, Zhang obtain several different forms of Aleksandrov-Fenchel inequality. Us-ing the classical Aleksandrov-Fenchel inequality, we study the symmetric mixed homothetic deficit△i(K, L)(where△i(K, L)=Vi2(K, L)-Vi-1(K, L)Vi+1(K, L) is the mixed volum of K, L) and the Aleksandrov-Fenchel isoperimetric deficit△i(K)=Wi2(K)-Wi-1(K)Wi+1(K)(where Wi(K) i-th quermassintegral of K), and obtain several lower bounds of the isoperimetric deficit. As direct corollar-ies, we obtain the lower bounds of isoperimetric deficit expressed by the width function of project body in two-dimensional, three-dimensional Euclidean space. Finally, we obtain an upper bounds of isoperimetric deficit. The results are:
Theorem4.2.10Let K be a convex body in Rn and Wi(K) i-th quer-massintegral of K. Then where u∈Sn-1, u⊥is the linear subspace perpendicular to u, Ku is the orthogonal projection of a convex body K onto the linear subspace u⊥, ωd-1(Ku) is the mean width of Ku, and ωmax(Ku) and ωmin(Ku) are the maximum and minimum width of ω(Ku) over all u∈Sn-1.
Theorem4.3.2Let K be a convex body in Rn and Wi(K) i-th quermass-integral of K. If1≤i≤n-1, then△i(K)=Wi2(K)-Wi+1(K)Wi-1(K)≤Wi+1(K)Wi(K)(R(K, B)-r(K, B)), where r(K, B) and R(K, B) are the relative inradius and the relative circumradius of K with respect to B, respectively.
引文
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[3]Zhang G., Geometric inequalities and inclusion measures of convex bodies, Mathematika,41(1994),95-116.
[4]Zhou J., Plan Bonnesen-type inequalities, Acta. Math. Sinica, Chinese Se-ries,50 (2007), no.6,1397-1402.
[5]Ren D., Topics in Integral Geometry, World Scientific, Sigapore (1994).
[6]陈省身,陈维桓,微分几何讲义(第二版),北京：北京大学出版社,2001.
[7]Santalo L. A., Integral Geometry and Geometric Probability, Reading, MA: Addison-Wesley,1976.
[8]Helgason S., Differential geometry, Lie groups, and symmetric spaces, U.S.A. Academic Press, INC.,1978.
[9]Banchoff T. F. and Pohl W. F., A generalization of the isoperimetric inequal-ity, J. Diff. Geo.,6 (1971),175-213.
[10]Bokowski J.& Heil E., Integral representation of quermassintegrals and Bonnesen-style inequalities, Arch. Math.,47 (1986),79-89.
[11]Bonnesen T., Les problems des isoperimetres et des isepiphanes, Gauthier-Villars, Paris (1929).
[12]Bonnesen, T., Les problemes des isoperimetres et des is epiphanes, Bull. Amer. Math. Soc.36(1930),
[13]Bonnesen T.& Fenchel W., Theorie der konvexen Koeper, Berlin-Heidelberg-New York (1934),2nd ed. (1974).
[14]O. Bottema, Eine obere Grenze fur das isoperimetrische Defizit ebener Kur-ven, Nederl. Akad. Wetensch. Proc., A66 (1933),442-446.
[15]Burago Y. D.& Zalgaller V. A., Geometric Inequalities, Springer-Verlag Berlin Heidelberg (1988).
[16]Diskant V., A generalization of Bonnesen's inequalities, Soviet Math. Dokl., 14 (1973),1728-1731 (Transl. of Dokl. Akad. Nauk SSSR,213(1973)).
[17]Grinberg E., Ren D. & Zhou J., The symetric isoperimetric deficit and the containment problem in a plan of constant curvature, preprint.
[18]Gysin L., The isoperimetric inequality for nonsimple closed curves, Proc. Amer. Math. Soc.,118 (1993), no.1,197-203.
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