Abstract
We settle J. Wetzel’s 1970’s conjecture and show that a \(30^{\circ }\) circular sector of unit radius can accommodate every planar arc of unit length. Leo Moser asked in 1966 for the (convex) region with the smallest area in the plane that can accommodate each arc of unit length. With area \(\pi /12,\) this sector is the smallest such set presently known. Moser’s question has prompted a multitude of papers on related problems over the past 50 years, most remaining unanswered.
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Acknowledgements
We are grateful to John E. Wetzel for providing the introductory material and the associated references, invaluable comments and guidance. Communication with Yevgenya Movshovich and her comments are very appreciated. We also appreciate the help from Banyat Sroysang and Pongbunthit Tonpho.We thank referee for helpful comments. The first author is supported by research fund of Mahidol University Internationl College. The second author is partially supported by the 90th Anniversary of Chulalongkorn University Fund (Ratchadaphiseksomphot Endowment Fund).
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Panraksa, C., Wichiramala, W. Wetzel’s sector covers unit arcs. Period Math Hung 82, 213–222 (2021). https://doi.org/10.1007/s10998-020-00354-x
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DOI: https://doi.org/10.1007/s10998-020-00354-x