关于等周型不等式的探索
Exploration on the Isoperimetric Inequalities
DOI: 10.12677/PM.2019.93043, PDF,    科研立项经费支持
作者: 陈 欣, 高 翔*：中国海洋大学数学科学学院，山东 青岛
关键词: 等周型不等式；Bonnesen-型不等式；积分几何；傅里叶级数；Isoperimetric Inequalities； Bonnesen-Type Inequalities； Integral Geometry； Fourier Series
摘要: 等周不等式是最古老的几何不等式之一并且被广泛地应用于学术研究和日常生活中。本文探索了等周型不等式的历史发展过程，以及近年来国内外关于等周型不等式的发展情况。此外，本文还对等周型不等式的未来发展趋势进行了合理的展望和探讨。
Abstract: The isometric inequality is one of the oldest geometric inequalities and is widely used in academic research and our daily life. This paper explores the historical development process of the iso-type inequalities and the development of the iso-type inequalities in recent years. In addition, this paper also makes a reasonable outlook and discussing on the future development trend of the iso-type inequalities.
文章引用：陈欣, 高翔. 关于等周型不等式的探索[J]. 理论数学, 2019, 9(3): 323-329. https://doi.org/10.12677/PM.2019.93043

参考文献

[1] 匡继昌. 一般不等式研究在中国的新进展[J]. 北京联合大学学报(自然科学版), 2005, 19(1): 29-37.
[2] 胡方杰. 等周不等式初探[J]. 中学数学研究(华南师范大学版), 2018(15): 45-48.
[3] 朱建华, 赵微, 孟新柱. 等周问题求解新探索[J]. 大学数学, 2015, 31(2): 20-23.
[4] Steiner, J. (1842) Sur le maximum et le minimum des figures dans le plan, sur la sphere, et dans l’espace en general, I and II. Journal für die reine und angewandte Mathematik, 24, 93-152, 189-250.
[Google Scholar] [CrossRef] 
[5] Kazarinoff, N.D. (1986) Geometric Inequalities. Peking University Press, Bei-jing.
[6] 任德麟. 积分几何引论[M]. 上海: 上海科技技术出版社, 1988.
[7] Ren, D. (1992) Topics in Integral Geometry. World Scientific International Publisher, Singapore.
[8] Santalo, L.A. (1976) Integral Geometry and Geometric Probability. Addi-son-Wesley, Reading.
[9] Hsiung, C.C. (1997) A First Course in Differential Geometry. International Press, Somer-ville.
[10] Grinberg, E., Ren, D. and Zhou, J. (2000) The Symmetric Isoperimetric Deficit and the Containment Problem in a Plane of Constant Curvature.
[11] 周家足. 积分几何与等周不等式[J]. 贵州师范大学学报(自然科学版), 2002, 20(2): 1-3.
[12] 周家足, 任德麟. 从积分几何的观点看几何不等式[J]. 数学物理学报, 2010, 30(5): 1322-1339.
[13] 周家足. 平面Bonnesen型不等式[J]. 数学学报, 2007, 50(6): 1397-1402.
[14] 郑高峰, 周阳. 几个关于极坐标的Bonnesen型不等式[J]. 数学杂志, 2018, 38(6): 1119-1122.
[15] 马磊, 戴勇. 关于四面体的Bonnesen型等周不等式[J]. 数学的实践与认识, 2013, 43(3): 232-235.
[16] Hadwiger, H. (1948) Die isoperimetrische Ungleichung im Raum. Elemente der Mathematik, 3, 25-38.
[17] Burago, Y.D. and Zalgaller, V.A. (1988) Geometric Inequalities. Springer-Verlag, Berlin Heidelberg.
[Google Scholar] [CrossRef] 
[18] Osserman, R. (1978) The Isoperimetric Inequality. Bulletin of the American Mathematical Society, 84, 1182-1238.
[Google Scholar] [CrossRef] 
[19] Osserman, R. (1979) Bonnesen-Style Isoperimetric Inequalities. The American Mathematical Monthly, 86, 1-29.
[Google Scholar] [CrossRef] 
[20] Zhang, G. and Zhou, J. (2006) Containment Measures in Integral Ge-ometry, Integral Geometry and Convexity. World Scientific, Singapore, 153-168.
[Google Scholar] [CrossRef] 
[21] Zhang, G. (1994) Geometric Inequalities and Inclusion Measures of Convex Bodies. Mathematika, 41, 95-116.
[Google Scholar] [CrossRef] 
[22] Zhou, J., Du, Y. and Cheng, F. (2012) Some Bonnesen-Style Inequalities for Higher Dimensions. Acta Mathematica Sinica, English Series, 28, 2561-2568.
[23] Bottema, O. (1933) Eine obere Grenze fur das isoperimetrische Defizit ebener Kurven. Koninklijke Nederlandse Akademie van Wetenschappen Proceedings. Series A, 66, 442-446.
[24] Pleijel, A. (1955) On konvexa kurvor. Nordisk Matematisk Tidskrift, 3, 57-64.
[25] Pan, S. and Xu, H. (2008) Stability of a Reverse Isoperimetric Inequality. Journal of Mathematical Analysis and Applications, 350, 348-353.
[Google Scholar] [CrossRef] 
[26] Pan, S.L. and Zhang, H. (2007) A Reverse Isoperimetric Inequality for Convex Plane Curves. Beiträge zur Algebra und Geometrie, 48, 303-308.
[27] Pan, S.L., Tang, X.Y. and Wang, X.Y. (2010) A Refined Reverse Isoperimetric Inequality in the Plane. Mathematical Inequalities & Applications, 13, 329-338.
[Google Scholar] [CrossRef] 
[28] Howard, R. (1998) The Sharp Sobolev Inequality and the Banchoff-Pohl Inequality on Surfaces. Proceedings of the American Mathematical Society, 126, 2779-2787.
[29] Klain, D. (2007) Bonnesen-Type Inequalities for Surfaces of Constant Curvature. Advances in Applied Mathematics, 39, 143-154.
[Google Scholar] [CrossRef] 
[30] Gao, X. (2011) A Note on the Reverse Isoperimetric Inequality. Results in Mathematics, 59, 83-90.
[Google Scholar] [CrossRef] 
[31] Gao, X. (2011) A Note on the Isoperimetric Inequality and Its Stability. Journal of Mathematical Inequalities, 5, 371-381.
[Google Scholar] [CrossRef] 
[32] Gao, X. (2012) A New Reverse Isoperimetric Inequality and Its Stability. Mathematical Inequalities & Applications, 15, 733-743.
[Google Scholar] [CrossRef] 
[33] Gao, X. and Li, C.J. (2015) The Isoperimetric Inequality and Its Stability. Journal of Mathematical Inequalities, 9, 897-912.
[Google Scholar] [CrossRef] 
[34] 何刚, 姜德烁. 平面凸体的等周亏格的上界估计[J]. 数学杂志, 2013, 33(2): 349-353.
[35] 戴勇, 姚惠. 关于平面卵形区域的等周亏格上界的几点注记[J]. 数学的实践与认识, 2013, 43(1): 188-191.
[36] 戴勇, 吴现荣, 刘朝军. 几个与Ros等周亏格相关的逆Bonnesen型不等式[J]. 数学的实践与认识, 2016, 46(18): 193-196.
[37] 戴勇, 邓玲芳. 关于R~3中卵形区域的等周亏格的上界估计的注记[J]. 数学杂志, 2013, 33(1): 153-156.
[38] 张增乐, 罗淼, 陈方维. 平面上的新凸体与逆Bonnesen-型不等式[J]. 西南师范大学学报(自然科学版), 2015, 40(4): 27-30.
[39] Gage, M.E. (1983) An Isoperimetric Inequality with Applications to Curve Shortening. Duke Mathematical Journal, 50, 1225-1229.
[Google Scholar] [CrossRef] 
[40] 潘生亮, 唐学远, 汪小玉. Gage等周不等式的加强形式[J]. 数学年刊A辑(中文版), 2008, 29(3): 301-306.
[41] 周家足, 姜德烁, 李明, 陈方维. 超曲面的Ros定理[J]. 数学学报, 2009, 52(6): 1075-1084.
[42] 马磊. 关于平面凸多边形的一个Bonnesen型不等式[J]. 数学杂志, 2015, 35(1): 154-158.
[43] Xia, Y.W., Xu, W.X., Zhou, J.Z. and Zhu, B.C. (2013) Reverse Bonnesen Style Inequalities in a Surface of Constant Curvature. Science China (Mathematics), 56, 1143-1152.
[Google Scholar] [CrossRef] 
[44] Li, M. and Zhou, J. (2010) An Isoperimetric Deficit Upper Bound of the Convex Domain in a Surface of Constant Curvature. Science China (Mathematics), 53, 1941-1946.
[Google Scholar] [CrossRef] 
[45] 张增乐. 关于Gauss曲率星体的几何不等式及其相关问题[D]: [博士学位论文]. 重庆: 西南大学, 2018.

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