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The reverse isoperimetric inequality for convex plane curves through a length-preserving flow

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Abstract

By a length-preserving flow, we provide a new proof of a conjecture on the reverse isoperimetric inequality composed by Pan et al. (Math Inequal Appl 13:329–338, 2010), which states that if \(\gamma \) is a convex curve with length L and enclosed area A, then the best constant \(\varepsilon \) in the inequality

$$\begin{aligned} L^2\le 4\pi A+\varepsilon |{\tilde{A}}| \end{aligned}$$

is \(\pi \), where \({\tilde{A}}\) denotes the oriented area of the locus of its curvature centers.

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References

  1. Chavel, I.: Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  2. Chou, K.S., Zhu, X.P.: The Curve Shortening Problem. Chapman & Hall/CRC, Boca Raton (2001)

    Book  Google Scholar 

  3. Dergiades, N.: An elementary proof of the isoperimetric inequality. Forum Geom. 2, 129–130 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Fang, J.B.: A reverse isoperimetric inequality for embedded starshaped plane curves. Arch. Math. (Basel) 108, 621–624 (2017)

    Article  MathSciNet  Google Scholar 

  5. Gao, X.: A note on the reverse isoperimetric inequality. Results Math. 59, 83–90 (2011)

    Article  MathSciNet  Google Scholar 

  6. Ivaki, M.N.: Centro-affine curvature flows on centrally symmetric convex curves. Trans. Amer. Math. Soc. 366, 5671–5692 (2014)

    Article  MathSciNet  Google Scholar 

  7. Osserman, R.: The isoperimetric inequalities. Bull. Amer. Math. Soc. 84, 1182–1238 (1978)

    Article  MathSciNet  Google Scholar 

  8. Pan, S.L., Tang, X.Y., Wang, X.Y.: A refined reverse isoperimetric inequality in the plane. Math. Inequal. Appl. 13, 329–338 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Pan, S.L., Yang, J.N.: On a non-local perimeter-preserving curve evolution problem for convex plane curves. Manuscr. Math. 127, 469–484 (2008)

    Article  MathSciNet  Google Scholar 

  10. Pan, S.L., Zhang, H.: A reverse isoperimetric inequality for convex plane curves. Beiträge Algebra Geom. 48, 303–308 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Pankrashkin, K.: An inequality for the maximum curvature through a geometric flow. Arch. Math. (Basel) 105, 297–300 (2015)

    Article  MathSciNet  Google Scholar 

  12. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press, Cambridge (2014)

    MATH  Google Scholar 

  13. Süssmann, B.: Curve shortening and the Banchoff–Pohl inequality in symmetric Minkowski geometries. Ann. Global Anal. Geom. 29, 329–338 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Süssmann, B.: Isoperimetric inequalities for special classes of curves. Differ. Geom. Appl. 29, 1–6 (2011)

    Article  MathSciNet  Google Scholar 

  15. Stancu, A.: Centro-affine invariants for smooth convex bodies. Int. Math. Res. Not. 10, 2289–2320 (2012)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Science, Dalian Maritime University, Dalian, 116026, People’s Republic of China

    Yunlong Yang & Weiping Wu

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  1. Yunlong Yang
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  2. Weiping Wu
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Correspondence to Yunlong Yang.

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This work was supported by the Fundamental Research Funds for the Central Universities (Nos. 3132020172, 3132019177).

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Yang, Y., Wu, W. The reverse isoperimetric inequality for convex plane curves through a length-preserving flow. Arch. Math. 116, 107–113 (2021). https://doi.org/10.1007/s00013-020-01541-5

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  • DOI: https://doi.org/10.1007/s00013-020-01541-5

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