加强的反向等周不等式及其稳定性
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摘要
在几何中，大多几何不等式都有这样一个性质：当取一些特殊的凸体时，比如球，圆，椭球等，等号成立．如果某个凸体使得它所满足的不等式与等号成立相差很小时，这个凸体与使得等号成立的特殊的凸体有多大偏差?这就是几何不等式的稳定性问题，几何不等式的稳定性是凸几何，积分几何和凸分析中一个重要的研究课题，最初的研究在Minkovski和Bonnesen的工作中已有涉及，但是直到上世纪80年代才得到系统的研究，其中H．Groemer，V I．Diskant，B．Fuglede等人研究了很多几何不等式的稳定性．
本文主要研究了如下加强的反向等周不等式及其稳定性问题，即若γ是平面上严格凸的闭曲线，p(γ)为其周长，a(γ)为γ所围区域的面积，则有其中(a|~)(γ)是γ的曲率中心轨迹所围区域的有向面积，且等号成立时当且仪当γ是一个圆周．最后利用类似的办法讨论了[18]中提出的问题．
Most geometric inequalities concerning convex bodies have the property that the occurence of the equality sign characterizes geometrically significant objects, like balls , ellipsoids . This fact suggests the following stability problem: If for some convex body the given inequality is satisfied so that it is not very different from an equality what can be said about the deviation of the body from these objects? Problems of this kind appear already in the work of Minkowski and Bonnesen, but have been investigated more systematically since the 1980s.
This thesis deals with a strengthened reverse isoperimetric inequality and its stability properties, That is, for closed strictly convex plane curveγwith length p(γ) and area a{γ),where (a|~)(γ) denotes the oriented area of the domain enclosed by the locus of curvature centers ofγ, and the equality holds if and only ifγis a circle. Finally,we discuss the problem which is pointed out in article [18] using the method above mentioned.
本文主要研究了如下加强的反向等周不等式及其稳定性问题，即若γ是平面上严格凸的闭曲线，p(γ)为其周长，a(γ)为γ所围区域的面积，则有其中(a|~)(γ)是γ的曲率中心轨迹所围区域的有向面积，且等号成立时当且仪当γ是一个圆周．最后利用类似的办法讨论了[18]中提出的问题．
Most geometric inequalities concerning convex bodies have the property that the occurence of the equality sign characterizes geometrically significant objects, like balls , ellipsoids . This fact suggests the following stability problem: If for some convex body the given inequality is satisfied so that it is not very different from an equality what can be said about the deviation of the body from these objects? Problems of this kind appear already in the work of Minkowski and Bonnesen, but have been investigated more systematically since the 1980s.
This thesis deals with a strengthened reverse isoperimetric inequality and its stability properties, That is, for closed strictly convex plane curveγwith length p(γ) and area a{γ),where (a|~)(γ) denotes the oriented area of the domain enclosed by the locus of curvature centers ofγ, and the equality holds if and only ifγis a circle. Finally,we discuss the problem which is pointed out in article [18] using the method above mentioned.
引文
[1] W. Blaschke, Kreis und Kugel, 2nd ed. Gruyter, Berlin, 1956.
[2] T. Bonnesen, W. Fenchel, Theorie der Convexen Korper, Chelsea Publishing, New York, 1948.
[3] K. J. Boroczky, Stability of the Blaschke-Santalo, and the affine isoperimetric inequalities, preprint, Oct. 2007.
[4] I. Chavel, Isoperimetric Inequalities, Differential Geometric and Analytic Perspectives, Combridge University Press, 2001.
[5] B. Fuglede, Stability in the isoperimetric problem, Bull. London Math. Soc. 18(1986), 599-605.
[6] R. J. Gardner, Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39(2002), 355-405.
[7] H. Groemer, Stability properties of geometric inequalities, Amer. Math. Monthly 97(1990), 382-394.
[8] H. Groemer, Stability properties of geometric inequalities, In: Handbook of Convex Geometry, P.M. Gruber and J.M. Wills (eds), North Holland, 125-150, 1993.
[9] H. Groemer, Geometric Applications of Fourier Series and Spherical Harmon- ics, Cambridge University Press, 1996.
[10] H. Groemer & R. Schneider, Stability estimates for some geometric inequlities, Bull. London Math. Soc. 23(1991), 67-74.
[11] M. Green & S. Osher, Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves, Asia J. Math. 3(1999), 659-676.
[12] B. Grunbaum, Convex Polytopes, John Wiley & Sons, London-New York-Sydney, 1967.
[13] C. C. Hsiung, A First Course in Differential Geometry, Pure Applied Math. Wiley, New York, 1981.
[14] R. Howard & A. Treigergs, A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature, Rocky Mountain J. Math. 25(1995), 635-684.
[15] A. Hurwitz, Sur quelques applications geometriques des series de Fourier, Ann. Ecole Norm. 19(1902), 357-408.
[16] R. Osserman, The isoperimetric inequalities, Bull. Amer. Math. Soc. 84(1978), 1182-1238.
[17] S. L. Pan, A note on the general plane cueve flows, J. Math. Study 33(2000), 17-26.
[18] S. L. Pan & H. Zhang, A reverse isoperimetric inequality for convex plane curves, Beitrage zur Algebra und Geoemtrie 48(2007), 303-308.
[19] R. Schneider, Kriimmungsschwerpunkte konvexer Korper I, Abh. Math. Sem. Univ. Hamburb 37(1972), 112-132.
[20] R. Schneider, Kriimmungsschwerpunkte konvexer Korper II, Abh. Math. Sem. Univ. Hamburb 37(1972), 204-217.
[21] R. Schneider, A stability estimate for the Aleksandrov-Fenchel inequality, with an application to mean curvature, Manus. Math. 69(1990), 291-300.
[22] R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, Cambridge University Press, Cambridge-New York, 1993.
[23] G. C. Shephard & R. R. Webster, Metrics for sets of convex bodies, Mathematika, 12(1965),73-88.
[24] J. Steiner, Sur le maximum et le minimum des figures dans le plan, sur la sphere, et dansl'espace en general, I and II, J. Reine Angew. Math. (Crelle), 24(1842), 93-152 and 189-250.
[25] G. Talenti, The standard isoperimetric inequality, in Handbook of Convex Geometry", Vol.A (edited by P. M. Gruber and J. M. Wills), pp.73-123, Amsterdam: North-Halland, 1993.
[26] R. A. Vitale, L_P metrics for compact convex sets, J. Approx. Theory 45(1985)280-287.
[2] T. Bonnesen, W. Fenchel, Theorie der Convexen Korper, Chelsea Publishing, New York, 1948.
[3] K. J. Boroczky, Stability of the Blaschke-Santalo, and the affine isoperimetric inequalities, preprint, Oct. 2007.
[4] I. Chavel, Isoperimetric Inequalities, Differential Geometric and Analytic Perspectives, Combridge University Press, 2001.
[5] B. Fuglede, Stability in the isoperimetric problem, Bull. London Math. Soc. 18(1986), 599-605.
[6] R. J. Gardner, Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39(2002), 355-405.
[7] H. Groemer, Stability properties of geometric inequalities, Amer. Math. Monthly 97(1990), 382-394.
[8] H. Groemer, Stability properties of geometric inequalities, In: Handbook of Convex Geometry, P.M. Gruber and J.M. Wills (eds), North Holland, 125-150, 1993.
[9] H. Groemer, Geometric Applications of Fourier Series and Spherical Harmon- ics, Cambridge University Press, 1996.
[10] H. Groemer & R. Schneider, Stability estimates for some geometric inequlities, Bull. London Math. Soc. 23(1991), 67-74.
[11] M. Green & S. Osher, Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves, Asia J. Math. 3(1999), 659-676.
[12] B. Grunbaum, Convex Polytopes, John Wiley & Sons, London-New York-Sydney, 1967.
[13] C. C. Hsiung, A First Course in Differential Geometry, Pure Applied Math. Wiley, New York, 1981.
[14] R. Howard & A. Treigergs, A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature, Rocky Mountain J. Math. 25(1995), 635-684.
[15] A. Hurwitz, Sur quelques applications geometriques des series de Fourier, Ann. Ecole Norm. 19(1902), 357-408.
[16] R. Osserman, The isoperimetric inequalities, Bull. Amer. Math. Soc. 84(1978), 1182-1238.
[17] S. L. Pan, A note on the general plane cueve flows, J. Math. Study 33(2000), 17-26.
[18] S. L. Pan & H. Zhang, A reverse isoperimetric inequality for convex plane curves, Beitrage zur Algebra und Geoemtrie 48(2007), 303-308.
[19] R. Schneider, Kriimmungsschwerpunkte konvexer Korper I, Abh. Math. Sem. Univ. Hamburb 37(1972), 112-132.
[20] R. Schneider, Kriimmungsschwerpunkte konvexer Korper II, Abh. Math. Sem. Univ. Hamburb 37(1972), 204-217.
[21] R. Schneider, A stability estimate for the Aleksandrov-Fenchel inequality, with an application to mean curvature, Manus. Math. 69(1990), 291-300.
[22] R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, Cambridge University Press, Cambridge-New York, 1993.
[23] G. C. Shephard & R. R. Webster, Metrics for sets of convex bodies, Mathematika, 12(1965),73-88.
[24] J. Steiner, Sur le maximum et le minimum des figures dans le plan, sur la sphere, et dansl'espace en general, I and II, J. Reine Angew. Math. (Crelle), 24(1842), 93-152 and 189-250.
[25] G. Talenti, The standard isoperimetric inequality, in Handbook of Convex Geometry", Vol.A (edited by P. M. Gruber and J. M. Wills), pp.73-123, Amsterdam: North-Halland, 1993.
[26] R. A. Vitale, L_P metrics for compact convex sets, J. Approx. Theory 45(1985)280-287.