关于计算凸域弦长分布函数的方法
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摘要
积分几何(Integral Geometry)起源于著名的Buffon投针问题,也称为几何概率(Geometric Probability),其实质就是通过各种积分来考察图形所具有的性质,因此本质上又属于微分几何.它的发展也始终和几何概率联系着.积分几何的思想是用概率的方法来研究凸体论及几何学。积分几何是数学的一门重要分支,它的应用和发展涉及代数学、偏微分方程、几何分析、凸几何、几何不等式等等。Santalo、严志达、吴大任、任德麟等前辈都为积分几何学做出了巨大的贡献。
凸体理论中另一个重要的课题是凸域的弦长分布函数,它有许多应用背景(模式识别、材料统计分析等),但迄今为止,现有文献中没有提供寻求凸域弦长分布函数的统一方法,本文以正三角形为例,讨论利用广义支持函数、限弦函数的概念来计算凸域的弦长分布函数的方法,该方法有普遍意义。
凸几何是以凸集或凸性作为研究对象的几何学分支,19世纪下,Hermann Brunn和Hermann Minkowski对凸几何学的早期发展做了大量开创性的工作。Brunn-Minkowski理论是凸几何学的经典内容,其核心部分是Brunn-Minkowski理论和混合体积理论。它和数学分支的很多重要方面都有深刻的联系。20世纪80年代,E.Lutwak引入了对偶混合体积的概念,进一步丰富了凸体这一理论,并由此解决了许多长期未能取得进展的重要问题。目前这部分内容仍然是凸几何中最为活跃的研究方向。20世纪90年代起,我国数学家张高勇在几何不等式、Busemann-Petty问题的终极性解答等方面取得了十分杰出的成果。
关于与仿射表面积有关的几何不等式也是重要的一方面,本文在张高勇老师得出的一些主要结论下讨论其中某些不等式的证明。
Integral Geometry is a subject which oringinated from the famous problem of Buffon needle.Another called of Integral Geometry is Geometric Probability, belonged to the category of differential geometry essentially, with the natures that investigating of a graphic through a variety of integral. The development has always been linked with geometric probability. The idea of Integral Geometry is to use the methods of probability to discuss convex body and geometry. The application and development have relationship with algebra,partial differential equation,geometric analysis,convex geometry,geometric inequality and so on.A lot of elderships made a great contribution to geometry,such as Santalo,Yan Zhida,Wu Daren,Ren Delin.
Another important subject of convex body theory is the chord length distribution function of convex domain.It has much application background such as Pattern Recognition,Statistical analysis of materials.But now ,there is not provide a unified approach to obtain the chord length distribution function for convex domain in existing literature.In this paper we take triangle for example to discuss the approach to calculate the exact analytical formula of chord length distribution function in using generalization support function technique and limited chord function. This method can also be used in other areas.
Convex Geometry is a branch of geometry,the study object is convex set or convex. In the 19th century, Hermann Brunn and Hermann Minkowski did a lot of pioneering work in the early development of the convex geometry. Brunn-Minkowski theory of convex geometry is a classic content .The core part is the Brunn-Minkowski theory and the theory of mixed volumes. Brunn-Minkowski theory has a profound link with many important branches of mathematics.In 1980s, E. Lutwak introduced the concept of dual mixed volume,further enriched the theory of convex bodies.And thus this solved many imporant issues which lacked of progress in a long term. At present this part of the convex geometry is still the most active research. Since the 1990s, China's mathematician Zhang Gao-Yong, has made a very excellent results in geometry inequality and ultimate solution of the Busemann-Petty problem.
The geometric inequality which related with affine surface area is also important on the one hand.This paper discuss the proof in some of the inequalities under the existing finding of Zhang Gaoyong.
凸体理论中另一个重要的课题是凸域的弦长分布函数,它有许多应用背景(模式识别、材料统计分析等),但迄今为止,现有文献中没有提供寻求凸域弦长分布函数的统一方法,本文以正三角形为例,讨论利用广义支持函数、限弦函数的概念来计算凸域的弦长分布函数的方法,该方法有普遍意义。
凸几何是以凸集或凸性作为研究对象的几何学分支,19世纪下,Hermann Brunn和Hermann Minkowski对凸几何学的早期发展做了大量开创性的工作。Brunn-Minkowski理论是凸几何学的经典内容,其核心部分是Brunn-Minkowski理论和混合体积理论。它和数学分支的很多重要方面都有深刻的联系。20世纪80年代,E.Lutwak引入了对偶混合体积的概念,进一步丰富了凸体这一理论,并由此解决了许多长期未能取得进展的重要问题。目前这部分内容仍然是凸几何中最为活跃的研究方向。20世纪90年代起,我国数学家张高勇在几何不等式、Busemann-Petty问题的终极性解答等方面取得了十分杰出的成果。
关于与仿射表面积有关的几何不等式也是重要的一方面,本文在张高勇老师得出的一些主要结论下讨论其中某些不等式的证明。
Integral Geometry is a subject which oringinated from the famous problem of Buffon needle.Another called of Integral Geometry is Geometric Probability, belonged to the category of differential geometry essentially, with the natures that investigating of a graphic through a variety of integral. The development has always been linked with geometric probability. The idea of Integral Geometry is to use the methods of probability to discuss convex body and geometry. The application and development have relationship with algebra,partial differential equation,geometric analysis,convex geometry,geometric inequality and so on.A lot of elderships made a great contribution to geometry,such as Santalo,Yan Zhida,Wu Daren,Ren Delin.
Another important subject of convex body theory is the chord length distribution function of convex domain.It has much application background such as Pattern Recognition,Statistical analysis of materials.But now ,there is not provide a unified approach to obtain the chord length distribution function for convex domain in existing literature.In this paper we take triangle for example to discuss the approach to calculate the exact analytical formula of chord length distribution function in using generalization support function technique and limited chord function. This method can also be used in other areas.
Convex Geometry is a branch of geometry,the study object is convex set or convex. In the 19th century, Hermann Brunn and Hermann Minkowski did a lot of pioneering work in the early development of the convex geometry. Brunn-Minkowski theory of convex geometry is a classic content .The core part is the Brunn-Minkowski theory and the theory of mixed volumes. Brunn-Minkowski theory has a profound link with many important branches of mathematics.In 1980s, E. Lutwak introduced the concept of dual mixed volume,further enriched the theory of convex bodies.And thus this solved many imporant issues which lacked of progress in a long term. At present this part of the convex geometry is still the most active research. Since the 1990s, China's mathematician Zhang Gao-Yong, has made a very excellent results in geometry inequality and ultimate solution of the Busemann-Petty problem.
The geometric inequality which related with affine surface area is also important on the one hand.This paper discuss the proof in some of the inequalities under the existing finding of Zhang Gaoyong.
引文
[1]任德麟.积分几何学引论[M].上海:上海科学技术出版社,1988:3-5,71-73
[2] Ren De-lin. Topics in integral geometry [M]. Singapore: World Scientific, 1994:70-78
[3] L.A.Santalo积分几何与几何概率[M].吴大任,译.天津:南开大学出版社1991.40-89
[4]程鹏,李寿贵,许金华.凸域内两点间的平均距离[J].数学杂志.2008,28(1):57-60
[5]赵静,李德宜,王现美.凸域内的平均长度[J].数学杂志,2007,27(3):291-294.
[6] S.Campi and P.Gronchi,The Lp -Bwsemann-Petty centroid inequality,Msthquality, Mathematic 49(2002),128-141
[7] C.Petty,Isoperimetric problems,Proc.Conf.Convexity and Combinato- rial Geometry,Univ.of Oklahoma,Norman,1971,26-41
[8] R.Ossermannn.The isoperrimetric inequality.Bull.Amer.Math.Doc. 1978,84,1182-1238.
[9] R.J.Gardner.The Brunn-Minkowski inequality.Bull.Amer.Math.Soc. 2002, 39,355-405.
[10] W.Firey.p-means of convex bodies.Math.Scand. 1962, 10,17-27
[11] E.Lutwak.The Brunn-Minkowski-Firey theory 1:Mixed volumes and the Minkowski problem,J.Differential Geom.,1993, 38,131-150
[12] E.Lutwak.The Brunn-Minkowski-Firey theory 2:Affine and geominimal surface areas Adv. Math. 1996, 118,244-294
[13] GaoyongZhang.The affine Sobolev inequality,J.Diff.Geom 1999, 53,183-202
[14] E.Lutwak,D.Yang and G.Zhang,A new ellipsoid associated with a convex body,Duck Math.J. 2000, 104,375-390
[15]张高勇、黎荣泽.某些凸多边形内定长线段运动测度公式及其在几何概率中的应用.武汉钢铁学院学报, 1983.
[16]朱兴萍,李德宜,何琼.凸域的外平行集的包含测度[J],数学杂志,2007,579-582.
[17]李平,许金华,李寿贵.正多边形与平行网相交的Buffon概率,黄冈师范学院学报,2007,12-16.
[18]许金华,兰春霞,程鹏.圆域内两类三角形的运动测度[J],数学杂志,2008,299-302.
[19] Chai Fang,Li Shougui,Zhang Hongzhou.The inequalities on geometric quantities of planar convex sets[J]. Journal of mathematics,2011(5).
[20] P. R. Scott, A family of inequalities for convex sets, Bull. Austral. Math.Soc., 20 (1979), 237–245.
[21] P.R. Scott, Sets of constant width and inequalities, Quart. J. Math. Ox-fordSer. (2), 32 (1981), 345–348.
[22]李德宜,平面凸集的完全不等式组[J].数学杂志. 25(2005), 49-52.
[23] M. A. Hernandez Cifre, Optimizing the perimeter and the area of convex sets with fixeddiameter and circumradius, Arch. Math. 138(2003), 95-110.
[24] E.Lutwak,D.Yang and G.Zhang,Sharp affine L pSobolev inequalities,J.Differential Georm.62(2002),17-38.
[25] E.Lutwak,D.Yang and G.Zhang, L p John ellipsoids,Proc.London Math.Soc.90(2005),497-520.
[26] Eric Grinberg,Delin Ren & Jiazu Zhou.the Symmetric Isoperimetric Deficit and the Containment Problem in a Plane of constant curvature,Preprint,2000.
[27] C.C.Hsiung.A First Courses in Differential Geometry[M].MA: International Press,1997.
[28] Steven R.Lay Convex sets and their applications[M]A Wiley interscience publication 1982.
[29]周家足;积分几何与等周不等式.贵州师范大学学报(自然科学版) 2(2002).
[30] E.Lutwak,J.Xiao,D.Yang and G.Zhang,Sharp L p isoperimetric inequalities for star bodies and their analytic inequalities,preprint.
[31] E.Lutwak,D.Yang and G.Zhang, Optimal Sobolev norms and the L p Minkowski problem,Int.Math.Res.Not.,2006,No.1,1-21.
[32] E.Lutwak,D.Yang and G.Zhang,A new ellipsoid associated with a convex body ,Duke Math.J.104(2000),375-132.
[33] H.Busemann,The isoperimetric problem for Minkowski area,Amer.J. Math.71(1949),743-762.
[34] M.Sholander, On certain minimum problems in the theory of convex curves, Trans. Amer. Math. Soc., 73 (1952), 139–173.
[35] R.Osserman, The isoperimtric inequality, Bull.Amer.Math.Soc. 84(6)(1978), 1182-1238.
[36] B.Andrews,Guss curvature flows:the fate of the rolling stones, Invent.Math.1389(1999),151-161.
[37] K.Ball,Volume ratios and a reverse isoperimetric inequality, J.London Math.Soc.44(1991),351-159.
[38] K.Chou and X.Wang,The L p Minkowski problem and the Minkowski problem in centroaffine geometry,Adv.Math.205(2006),33-83.
[39] R.J.Gardner,Geometric Tomography,2nd Ed.,Cambridge University Press Cambridge,2006.
[40] S.Campi and P.Gronchi,The L p Busemann Petty centroid inequality, Adv.Math.167(2002),128-141.
[41] S.Campi and P.Gronchi,On the reverse L p Busemann Petty centroid inequality,Mathematika 49(2002),1-11.
[42] W.Chen, L p Minkowski problem with not necessarily positive data,Adv. Math.2019(2006),77-89.
[43] R.Ossermann,The isoperimetric inequality,Bull.Amer.Math.Soc. 84(1987),1182-1238.
[2] Ren De-lin. Topics in integral geometry [M]. Singapore: World Scientific, 1994:70-78
[3] L.A.Santalo积分几何与几何概率[M].吴大任,译.天津:南开大学出版社1991.40-89
[4]程鹏,李寿贵,许金华.凸域内两点间的平均距离[J].数学杂志.2008,28(1):57-60
[5]赵静,李德宜,王现美.凸域内的平均长度[J].数学杂志,2007,27(3):291-294.
[6] S.Campi and P.Gronchi,The Lp -Bwsemann-Petty centroid inequality,Msthquality, Mathematic 49(2002),128-141
[7] C.Petty,Isoperimetric problems,Proc.Conf.Convexity and Combinato- rial Geometry,Univ.of Oklahoma,Norman,1971,26-41
[8] R.Ossermannn.The isoperrimetric inequality.Bull.Amer.Math.Doc. 1978,84,1182-1238.
[9] R.J.Gardner.The Brunn-Minkowski inequality.Bull.Amer.Math.Soc. 2002, 39,355-405.
[10] W.Firey.p-means of convex bodies.Math.Scand. 1962, 10,17-27
[11] E.Lutwak.The Brunn-Minkowski-Firey theory 1:Mixed volumes and the Minkowski problem,J.Differential Geom.,1993, 38,131-150
[12] E.Lutwak.The Brunn-Minkowski-Firey theory 2:Affine and geominimal surface areas Adv. Math. 1996, 118,244-294
[13] GaoyongZhang.The affine Sobolev inequality,J.Diff.Geom 1999, 53,183-202
[14] E.Lutwak,D.Yang and G.Zhang,A new ellipsoid associated with a convex body,Duck Math.J. 2000, 104,375-390
[15]张高勇、黎荣泽.某些凸多边形内定长线段运动测度公式及其在几何概率中的应用.武汉钢铁学院学报, 1983.
[16]朱兴萍,李德宜,何琼.凸域的外平行集的包含测度[J],数学杂志,2007,579-582.
[17]李平,许金华,李寿贵.正多边形与平行网相交的Buffon概率,黄冈师范学院学报,2007,12-16.
[18]许金华,兰春霞,程鹏.圆域内两类三角形的运动测度[J],数学杂志,2008,299-302.
[19] Chai Fang,Li Shougui,Zhang Hongzhou.The inequalities on geometric quantities of planar convex sets[J]. Journal of mathematics,2011(5).
[20] P. R. Scott, A family of inequalities for convex sets, Bull. Austral. Math.Soc., 20 (1979), 237–245.
[21] P.R. Scott, Sets of constant width and inequalities, Quart. J. Math. Ox-fordSer. (2), 32 (1981), 345–348.
[22]李德宜,平面凸集的完全不等式组[J].数学杂志. 25(2005), 49-52.
[23] M. A. Hernandez Cifre, Optimizing the perimeter and the area of convex sets with fixeddiameter and circumradius, Arch. Math. 138(2003), 95-110.
[24] E.Lutwak,D.Yang and G.Zhang,Sharp affine L pSobolev inequalities,J.Differential Georm.62(2002),17-38.
[25] E.Lutwak,D.Yang and G.Zhang, L p John ellipsoids,Proc.London Math.Soc.90(2005),497-520.
[26] Eric Grinberg,Delin Ren & Jiazu Zhou.the Symmetric Isoperimetric Deficit and the Containment Problem in a Plane of constant curvature,Preprint,2000.
[27] C.C.Hsiung.A First Courses in Differential Geometry[M].MA: International Press,1997.
[28] Steven R.Lay Convex sets and their applications[M]A Wiley interscience publication 1982.
[29]周家足;积分几何与等周不等式.贵州师范大学学报(自然科学版) 2(2002).
[30] E.Lutwak,J.Xiao,D.Yang and G.Zhang,Sharp L p isoperimetric inequalities for star bodies and their analytic inequalities,preprint.
[31] E.Lutwak,D.Yang and G.Zhang, Optimal Sobolev norms and the L p Minkowski problem,Int.Math.Res.Not.,2006,No.1,1-21.
[32] E.Lutwak,D.Yang and G.Zhang,A new ellipsoid associated with a convex body ,Duke Math.J.104(2000),375-132.
[33] H.Busemann,The isoperimetric problem for Minkowski area,Amer.J. Math.71(1949),743-762.
[34] M.Sholander, On certain minimum problems in the theory of convex curves, Trans. Amer. Math. Soc., 73 (1952), 139–173.
[35] R.Osserman, The isoperimtric inequality, Bull.Amer.Math.Soc. 84(6)(1978), 1182-1238.
[36] B.Andrews,Guss curvature flows:the fate of the rolling stones, Invent.Math.1389(1999),151-161.
[37] K.Ball,Volume ratios and a reverse isoperimetric inequality, J.London Math.Soc.44(1991),351-159.
[38] K.Chou and X.Wang,The L p Minkowski problem and the Minkowski problem in centroaffine geometry,Adv.Math.205(2006),33-83.
[39] R.J.Gardner,Geometric Tomography,2nd Ed.,Cambridge University Press Cambridge,2006.
[40] S.Campi and P.Gronchi,The L p Busemann Petty centroid inequality, Adv.Math.167(2002),128-141.
[41] S.Campi and P.Gronchi,On the reverse L p Busemann Petty centroid inequality,Mathematika 49(2002),1-11.
[42] W.Chen, L p Minkowski problem with not necessarily positive data,Adv. Math.2019(2006),77-89.
[43] R.Ossermann,The isoperimetric inequality,Bull.Amer.Math.Soc. 84(1987),1182-1238.