Abstract
For all unit radius spherical convex k-gons of a fixed diameter \(\pi /2\), where \(3\le k\le n\) and n is odd, the regular spherical triangle has the maximal perimeter, and the regular spherical n-gon attains the maximal area. Besides, for all unit radius spherical convex k-gons of a fixed thickness \(\pi /2\), where \(3\le k\le n\) and n is odd, the regular spherical n-gon attains the minimum perimeter, and the regular triangle reaches the minimal area. Particularly, the reduced spherical polygons reach the maximal thickness (resp. minimal diameter) \(\pi /2\) among all spherical convex polygons of fixed diameter (resp. thickness) \(\pi /2\). Moreover, when n is odd or even, a few conclusions of unit radius spherical convex n-gons are established, respectively.
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The last part of the paper was added according to the suggestion of an anonymous reviewer. The authors would like to thank the referees for their many valuable comments and suggestions that helped us to improve the quality of the paper.
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Liu, C., Chang, Y. Extremal problems for spherical convex polygons. Arch. Math. 118, 435–450 (2022). https://doi.org/10.1007/s00013-021-01698-7
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DOI: https://doi.org/10.1007/s00013-021-01698-7