Abstract
Let K be a convex body in \({\mathbb {R}} ^d\), with \(d = 2,3\). We determine sharp sufficient conditions for a set E composed of 1, 2, or 3 points of \(\mathrm{bd}K\), to contain at least one endpoint of a diameter of K. We extend this also to convex surfaces, with their intrinsic metric. Our conditions are upper bounds on the sum of the complete angles at the points in E. We also show that such criteria do not exist for \(n\ge 4\) points.
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Acknowledgements
The authors would like to thank Professor Joseph O’Rourke for his valuable comments. The first author was partially supported by a Grant-in-Aid for Scientific Research (C) (No. 17K05222), Japan Society for Promotion of Science. The last two authors gratefully acknowledge financial support by NSF of China (11871192, 11471095). The last three authors direct their thanks to the Program for Foreign Experts of Hebei Province (No. 2019YX002A). The research of the last author was also partly supported by the International Network GDRI ECO-Math.
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Itoh, Ji., Vîlcu, C., Yuan, L. et al. Locating Diametral Points. Results Math 75, 68 (2020). https://doi.org/10.1007/s00025-020-01193-5
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DOI: https://doi.org/10.1007/s00025-020-01193-5