逆向等周型不等式的研究进展
Advances in Inverse Isoperimetric Inequalities
摘要: 数学中最经典的几何不等式就是等周不等式，它刻画了欧式平面中的由简单闭曲线所围区域的面积与周长之间的关系。本文从最经典的等周不等式出发，探究逆向等周不等式的发展进程并归纳总结近年来逆向等周不等式的研究成果，主要从三个方面分别介绍了逆向等周不等式在平面卵形域、高维欧式曲面、流行曲面及一些特殊曲面的发展过程及主要研究成果。
Abstract: The most classical geometric inequality in mathematics is the isoperimetric inequality, which de-scribes the relationship between the area and perimeter of the region enclosed by a simple close curve in a Euclidean plane. Starting from the most classical isoperimetric inequalities, this paper explores the development process of reverse isoperimetric inequalities and summarizes the research results of reverse isoperimetric inequalities in recent years. It mainly introduces the development process and main research results of reverse isoperimetric inequalities in plane oval domain, high-dimensional Euclidean surface, popular surface and some special surface from three aspects.
文章引用：贾艳丽, 高翔. 逆向等周型不等式的研究进展[J]. 理论数学, 2019, 9(2): 152-163. https://doi.org/10.12677/PM.2019.92020

参考文献

[1] Bokowski, J. and Heil, E. (1986) Integral Representation of Quermassintegrals and Bonnesen-Style Inequalities. Archiv der Mathematik, 47, 79-89.
[Google Scholar] [CrossRef] 
[2] 周家足. 平面Bonnesen型不等式[J]. 数学学报, 2007, 50(6): 1397-1402.
[3] Diskant, V. (1973) A Generalization of Bonnesen’s Inequalities. Soviet Mathematics Doklady, 14, 1728-1732.
[4] Zhang, X.-M. (1997) Bonessen-Style Inequality and Pseudo-Perimeters for Polygons. Journal of Geometry, 60, 188-201.
[Google Scholar] [CrossRef] 
[5] 增春娜, 周家足, 岳双珊. 平面两凸域的对称混合等周不等式[J]. 数学学报, 2012, 5(2): 355-362.
[6] Osserman, R. (1979) The Bonessen-Style Isoperimetric Inequality. The American Mathematical Monthly, 86, 1-29.
[Google Scholar] [CrossRef] 
[7] Zhou, J. and Cheng, Y. (2012) Some Bonessen-Style Inequality for Higher Dimensions. Acta Mathematica Sinica, 28, 2561-2568.
[Google Scholar] [CrossRef] 
[8] Bottema, O. (1933) Eineobere Grenze fur das isoperimetrische Defizitebener Kurven. Nederl Akad Wetensch Proc Ser A, 66, 442-446.
[9] Pleijiel, A. (1955) On konvexa kurvor. Nordisk Math Tidskr, 3, 57-64.
[10] Pan, S.L. and Zhang, H. (2007) A Reverse Isoperimetric Inequality for Convex Plane Curves. Beitragezur Algebra und Geometrie Contributions to Algebra and Geometry, 48, 303-308.
[11] Gao, X. (2011) A Note on the Reverse Isoperimetric Inequality. Results in Mathematics, 59, 83-90.
[Google Scholar] [CrossRef] 
[12] Pan, S.L. and Yang, J.N. (2008) On a Non-Local Perimeter-Preserving Curve Evolution Problem for Convex Plane Curves. Manuscripta Mathematica, 127, 469-484.
[Google Scholar] [CrossRef] 
[13] Howard, R. and Treibergs, A. (1995) A Reverse Isoperimetric Inequality, Stability and Extremal Theorems for Plane Curves with Bounded Curvature. Rocky Mountain Journal of Mathematics, 25, 635-684.
[Google Scholar] [CrossRef] 
[14] Li, C.J. and Gao, X. (2015) The Isoperimetric Inequality and Its Stability. Journal of Mathematics, 3, 897-912.
[15] Santalo, L. (1942) Integral Formulas in Crofton’s Style on the Sphere and Some Inequalities Referring to Spherical Curves. Duke Mathematical, 9, 707-722.
[Google Scholar] [CrossRef] 
[16] 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): 1322-1339.
[17] Zhou, J.Z., Ma, L. and Xu, W. (2013) On the Isoperimetric Dedicit Upper Limit. Bulletin of the Korean Mathematical Society, 50, 175-184.
[Google Scholar] [CrossRef] 
[18] 张洪, 罗永超, 徐文学. Bonnesen型Ros等周不等式[J]. 数学的实践与认识, 2015, 45(17): 263-266.
[19] 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.
[20] Li, M. and Zhou, J.Z. (2010) An Upper Limit for the Isoperimetric Deficit of Convex Set in Plane of Constant Curvature. SCI China Math, 53, 1941-1946.
[Google Scholar] [CrossRef] 
[21] Xia, Y. and Xu, W. (2013) Reverse Bonnesen Style Inequalities in a Surface of Constant Curvature. SCI China Math, 6, 1145-4454.
[Google Scholar] [CrossRef] 
[22] Zhou, J. and Ren, D. (2010) Geometric Inequalities—From Integral Geometry Point of View. Acta Mathematica Scientia, 30, 1322-1339.
[23] Fang, J. (2017) A Reverse Isoperimetric Inequality for Embedded Starshaped Plane Curves. Archiv der Mathematik, 108, 621-624.
[Google Scholar] [CrossRef] 
[24] Alexandrov, A.D. (1945) One Isoperimetric Inequality Problem. Doklady Akademii Nauk SSSR, 50, 31-34.
[25] Borisenko, A. (2016) Reverse Isoperimetric Inequality in Two-Dimensional Alexandrov Spaces.
[26] Green, M. and Osher, S. (1999) Steiner Polynomials, Wulff Flows, and Some New Isoperimetric Inequalities for Convex Plane Curves. Asian Journal of Mathematics, 3, 659-676.
[Google Scholar] [CrossRef] 
[27] Zhang, Z.L. and Zhou, J.Z. (2017) Bonnesen-Style Wulff Is Operimetric Inequality. Journal of Inequalities and Applications, 2017, 42.
[Google Scholar] [CrossRef] [PubMed]
[28] 王鹏富, 徐文学, 周家足, 朱促成. 平面两凸域的Bonnesen型对称混合不等式[J]. 中国科学(数学), 2015(45): 245-254.
[29] 王鹏富. Bonnesen型对称混合不等式[R]. 重庆市: 西南大学, 2016: 45-54.
[30] 罗淼. Bonnesen型对称混合等拟不等式与 混合质心体[R]. 重庆市: 西南大学, 2016: 56-69.
[31] Luo, M., Xu, W. and Zhou, J. (2015) Translative Containment Measure and Symmetric Moxed Isohomothetic Inequalities. Science China Mathematics, 58, 2593-3610.
[Google Scholar] [CrossRef] 

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