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数学与物理
理论数学
Vol. 13 No. 1 (January 2023)
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平面上保长度的闭凸曲线流及其应用
Closed-Convex Curve Flow with Length Preservation on the Plane and Its Application
DOI:
10.12677/PM.2023.131005
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PDF
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,
,
被引量
作者:
张永志
,
李亚尊
:云南师范大学,云南 昆明
关键词:
保长度流
;
常宽曲线
;
几何不等式
;
Preserve Length Flow
;
Constant Width Curve
;
Geometric Inequality
摘要:
该文研究了平面上保长度的闭凸曲线流及其应用,在这篇文章中,给出了如果初始闭凸曲线是宽度的常宽曲线,那么在该流下发展曲线保持常宽,并且宽度与初始曲线宽度相等,并利用该曲线流证明平面上闭凸曲线的一个曲率倒数积分的几何不等式,对等号成立做了详细的几何分类解释。
Abstract:
In this paper, the closed-convex curve flow with preserved length on the plane and its application are studied. If the initial closed-convex curve is a constant width curve of width, then the development curve remains constant width under the flow, and the width is equal to the width of the initial curve, and the curve flow is used to prove the geometric inequality of the reciprocal integral of a curvature of the closed convex curve on the plane, and the equivalence sign is established to explain the geometric classification in detail.
文章引用:
张永志, 李亚尊. 平面上保长度的闭凸曲线流及其应用[J]. 理论数学, 2023, 13(1): 46-54.
https://doi.org/10.12677/PM.2023.131005
参考文献
[1]
潘生亮.几何不等式与曲率流[D]: [博士学位论文].上海: 华东师范大学,2001.
[2]
Yang, Y.L. (2020) An Inequality for the Minimum Affine Curvature of a Plane Curve. Comptes Rendus Mathematique, 358, 139-142.
https://doi.org/10.5802/crmath.19
[3]
Yang, Y. and Wu, W. (2021) The Reverse Isoperimetric Inequality for Convex Plane Curves through a Length-Preserving Flow. Archiv der Mathematik, 116, 107-113.
https://doi.org/10.1007/s00013-020-01541-5
[4]
夏康杰,郭洪欣.曲线流在平面曲线的几个不等式中的应用[J/OL].数学学报(中文版), 2022.
https://kns.cnki.net/kems/detail/11.2038.01.20220318.1020.014.html,2022-03-21
.
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梅向明,黄敬之.微分几何[M].北京: 北京高等教育出版社,2008.
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任德麟.积分几何学引论[M].上海: 上海科学技术出版社,1988.
[7]
Gao, L., Zhang, Z. and Zhou, F. (2020) An Extension of Rabinowitz's Polynomial Representation for Convex Curves. Beitrage zur Algebra und Geometrie, 61, 455-464.
https://doi.org/10.1007/s13366-020-00494-8
[8]
Li, C.J. and Gao, X. (2015) The Isoperimetric Inequality and Its Stability. Journal of Mathematics, 3, 897-912.
https://doi.org/10.7153/jmi-09-74
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