Abstract
The main result in this article is the following: Let \(K\subset \mathbb R^2\) be a regular convex body and let \(\alpha \), \(\beta \), \(\theta \), be three angles such that K has \(\alpha \)-chords, \(\beta \)-chords, and \(\theta \)-chords of constant length and \(\alpha +\beta +\theta =\pi \), then K is a disc. We also prove another characterization of the disc with respect to properties of its \((\alpha ,\beta ,\theta )\)-circumscribed triangles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chakerian, G.D., Groemer, H.: Convex bodies of constant width. In: Gruber, P.M., Wills, J.M. (eds.) Convexity and Its Applications. Birkhäuser, Basel (1983)
Cieślak, W., Miernowski, A., Mozgawa, W.: Isoptics of a closed strictly convex curve. Lect. Notes Math. 1481, 28–35 (1991)
Groemer, H.: Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge Univ. Press, Cambridge (1996)
Jerónimo-Castro, J.: A characterization of the disc by the angle of the support cone, Results Math. 76, no. 3, Paper No. 130, 9 pp (2021)
Jerónimo-Castro, J., Yee-Romero, C.: An inequality for the length of isoptic chords of convex bodies. Aequat. Math. 93(3), 619–628 (2019)
Jerónimo-Castro, J., Jimenez-Lopez, F.G.: Symmetries of convex sets in the hyperbolic plane. J. Conv. Anal. 26, 1077–1088 (2019)
Jerónimo-Castro, J., Jimenez-Lopez, F.G., Jiménez-Sánchez, R.: On convex bodies with isoptic chords of constant length. Aequ. Math. 94(6), 1189–1199 (2020)
Jerónimo-Castro, J., Rojas-Tapia, M.A., Velasco-García, U., Yee-Romero, C.: An isoperimetric type inequality for isoptic curves of convex bodies. Results Math. 75, 134 (2020)
Jerónimo-Castro, J., Jimenez-Lopez, F.G., Velasco-García, U.: Some characterizations of the circle related to circumscribed equiangular polygons. Bol. Soc. Mat. Mex. 27(3), 8 (2021)
Valentine, F.A.: Convex Sets. McGraw-Hill Series in Higher Mathematics, McGraw-Hill, New York (1964)
Yaglom, I.M., Boltyanski, V.G.: Convex Figures. Holt, Rinehart and Winston, London (1961)
Author information
Authors and Affiliations
Contributions
JJ and FJ wrote the manuscript and JJ produce the figures. All authors reviewed the manuscript
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ayala-Figueroa, R.I., González-García, I., Jerónimo-Castro, J. et al. Some characterizations of the disc by properties of isoptic triangles. Aequat. Math. 98, 591–602 (2024). https://doi.org/10.1007/s00010-023-00983-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-023-00983-w