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The Lower Bounds of the Mixed Isoperimetric Deficit

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Abstract

Let \(K_i\) be plane convex bodies with the perimeters \(L_{K_i}\) and areas \(A_{K_i}\) for \(i=1,2\), respectively. In this paper, the lower bounds of the mixed isoperimetric deficit \(\Delta _{K_1,K_2}=L_{K_1}^2L_{K_2}^2-16\pi ^2 A_{K_1}A_{K_2}\) are obtained; these bounds involve the areas enclosed by the evolute and the Wigner caustic, Hausdorff and \(L_2\) distances between two convex bodies.

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Acknowledgements

The author would like to thank the referees for their careful reading of the manuscript of the paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Huaibei Normal University, Huaibei, 235000, People’s Republic of China

    Deyan Zhang

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  1. Deyan Zhang
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Correspondence to Deyan Zhang.

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Communicated by Rosihan M. Ali.

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This work was supported by the Natural Science Foundation of Anhui Province (No. 1908085MA05) and University Natural Science Research Project of Anhui Province (No. KJ2019A0590) and Excellent Young Talents Fund Program of Higher Education Institutions of Anhui Province (gxyqZD2020022).

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Zhang, D. The Lower Bounds of the Mixed Isoperimetric Deficit. Bull. Malays. Math. Sci. Soc. 44, 2863–2872 (2021). https://doi.org/10.1007/s40840-020-01067-7

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  • DOI: https://doi.org/10.1007/s40840-020-01067-7

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