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Locating Diametral Points

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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Correspondence to Liping Yuan.

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Itoh, J., 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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Keywords

  • Convex body
  • diameter
  • geodesic diameter
  • diametral point

Mathematics Subject Classification

  • 52A10
  • 52A15
  • 53C45