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The Perimeter and Area of Reduced Spherical Polygons of Thickness \(\pi /2\)

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Abstract

We confirm a few recent conjectures of Lassak on the perimeter and area of reduced spherical polygons of thickness \(\pi /2\). This paper is based on the study of the sufficient and necessary conditions whether a spherical polygon of thickness \(\pi /2\) is reduced. The perimeter (resp. area) of every reduced spherical k-gon of thickness \(\pi /2\) is not greater than that of the regular spherical triangle (resp. regular spherical n-gon) of thickness \(\pi /2\), where \(3\le k\le n\). Moreover, the regular spherical odd-gon with n vertices and thickness \(\pi /2\) has the minimum perimeter (resp. maximum area) among all reduced spherical k-gons of thickness \(\pi /2\), where \(3\le k\le n\).

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Acknowledgements

The authors would like to thank the anonymous referees for their many valuable comments and suggestions that helped to improve the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China

    Yanxun Chang & Cen Liu

  2. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China

    Zhanjun Su

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  1. Yanxun Chang
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  2. Cen Liu
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  3. Zhanjun Su
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Correspondence to Yanxun Chang.

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Yanxun Chang: Supported by NSFC under Grant 11971053. Zhanjun Su: Science Foundation of Hebei Normal University (L2020Z01)

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Chang, Y., Liu, C. & Su, Z. The Perimeter and Area of Reduced Spherical Polygons of Thickness \(\pi /2\). Results Math 75, 135 (2020). https://doi.org/10.1007/s00025-020-01263-8

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  • DOI: https://doi.org/10.1007/s00025-020-01263-8

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