Abstract
In this note we consider a family of flows of convex and closed plane curves which interpolates the area-preserving and the length-preserving flows of Tsai and Wang. When the initial curve is convex, the flow exists for all \(t\in [0,\infty )\) and the convexity is preserved. The length is non-increasing while the enclosed area is non-decreasing during the process. When time goes to infinity, the isoperimetric deficit converges to 0 exponentially fast. As a consequence, the ratio of the inradius and circumradius converges to 1 exponentially fast.
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Acknowledgements
We are very grateful to the referee for his careful reading, expertise suggestions and helpful corrections which we have followed to improve the presentation of our results significantly.
Funding
Supported by the National Natural Science Foundation of China (grant no. 11971355) and Zhejiang Provincial Natural Science Foundation of China (grant no. LY22A010007).
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Communicated by Joachim Escher.
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Guo, H., Zhu, S. Linear interpolation on \(k^{\alpha }\)-type area-preserving and length-preserving curve flows. Monatsh Math 200, 81–91 (2023). https://doi.org/10.1007/s00605-022-01809-8
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DOI: https://doi.org/10.1007/s00605-022-01809-8