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An inequality for the length of isoptic chords of convex bodies

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Abstract

In this note we prove the following result: for every \(\alpha \in (0,\pi )\) and for a given convex body K in the plane, with minimal width w, there exists a chord [x, y] with length larger than or equal to \(w\cos \frac{\alpha }{2}\) such that there are support lines of K through x and y which form an angle \(\alpha .\) Moreover, if there is no such chord with length exceeding \(w\cos \frac{\alpha }{2}\), then K is a Euclidean disc.

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Acknowledgements

We thank the unknown referee for many valuable suggestions which improve the exposition of the paper.

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Authors and Affiliations

  1. Universidad Autónoma de Querétaro, Querétaro, Mexico

    Jesús Jerónimo-Castro

  2. Universidad Autónoma de Baja California, Ensenada, Baja California, Mexico

    Carlos Yee-Romero

Authors
  1. Jesús Jerónimo-Castro
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  2. Carlos Yee-Romero
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Correspondence to Jesús Jerónimo-Castro.

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Jerónimo-Castro, J., Yee-Romero, C. An inequality for the length of isoptic chords of convex bodies. Aequat. Math. 93, 619–628 (2019). https://doi.org/10.1007/s00010-018-0611-2

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  • DOI: https://doi.org/10.1007/s00010-018-0611-2

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