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Borsuk Number for Planar Convex Bodies

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Abstract

By using some simple tools from graph theory, we obtain a characterization of the compact sets in \(\mathbb {R}^n\) with Borsuk number equal to two. This result allows to give some examples of planar (convex) compact sets with Borsuk number equal to three. Moreover, we also prove that the unique centrally symmetric planar convex compact sets with Borsuk number equal to three are the Euclidean balls.

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Acknowledgements

The authors would like to thank the referee, for several helpful comments, Instituto de Matemáticas de la Universidad de Sevilla (IMUS), where this work was initiated, and Salvador Segura Gomis and José Pedro Moreno, for useful discussions.

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  1. Universidad de Sevilla, Seville, Spain

    Antonio Cañete

  2. University of Applied Sciences Zittau/Görlitz, Zittau, Germany

    Uwe Schnell

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  1. Antonio Cañete
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  2. Uwe Schnell
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Correspondence to Antonio Cañete.

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Antonio Cañete is partially supported by the MICINN project MTM2017-84851-C2-1-P, and by Junta de Andalucía Grant FQM-325.

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Cañete, A., Schnell, U. Borsuk Number for Planar Convex Bodies. Results Math 75, 14 (2020). https://doi.org/10.1007/s00025-019-1135-3

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