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Wetzel’s sector covers unit arcs

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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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References

  1. J.R. Alexander, J.E. Wetzel, W. Wichiramala, The \(\Lambda \)-property of a simple arc (2014) (Unpublished). https://doi.org/10.13140/RG.2.2.28756.55684

  2. P. Coulton, Y. Movshovich, Besicovitch triangles cover unit arcs. Geom. Dedicata 123, 79–88 (2006). https://doi.org/10.1007/s10711-006-9107-7

    Article  MathSciNet  MATH  Google Scholar 

  3. T. Khandhawit, D. Pagonakis, S. Sriswasdi, Lower Bound for Convex Hull Area and Universal Cover Problems. Int. J. Comput. Geom. Appl. 23(3), 197–212 (2013). https://doi.org/10.1142/S0218195913500076

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Moser, Poorly formulated unsolved problems in combinatorial geometry (1966) (mimeographed)

  5. W.O. Moser, Problems, problems, problems. Discrete Appl. Math. 31(2), 201–225 (1991). https://doi.org/10.1016/0166-218X(91)90071-4

    Article  MathSciNet  MATH  Google Scholar 

  6. Y. Movshovich, \(\Lambda \)-configurations and embeddings, to appear in J. Geom

  7. Y. Movshovich, J.E. Wetzel, Drapeable unit arcs fit in the unit \(30^{\circ }\) sector. to appear in Adv. Geom. 17(4), 497–506 (2017). https://doi.org/10.1515/advgeom-2017-0011

  8. R. Norwood, G. Poole, M. Laidacker, The worm problem of Leo Moser. Discrete Comput. Geom. 7(2), 153–162 (1992). https://doi.org/10.1007/BF02187832

    Article  MathSciNet  MATH  Google Scholar 

  9. W. Wang, An improved upper bound for worm problem. Acta Math. Sin. Chin. Ser. 49(4), 835–846 (2006). (doi: cnki:ISSN:0583-1431.0.2006-04-013)

    MathSciNet  MATH  Google Scholar 

  10. J.E. Wetzel, Fits and covers. Math. Mag. 76(5), 349–363 (2003). https://doi.org/10.1080/0025570X.2019.1523648

    Article  MathSciNet  MATH  Google Scholar 

  11. J.E. Wetzel, Sectorial covers for curves of constant length. Can. Math. Bull. 16, 367–375 (1973). https://doi.org/10.4153/CMB-1973-058-8

    Article  MathSciNet  MATH  Google Scholar 

  12. J.E. Wetzel, Bounds for covers of unit arcs. Geombinatorics XXII(3), 116–122 (2013)

    MathSciNet  MATH  Google Scholar 

  13. J.E. Wetzel, W. Wichiramala, A covering theorem for families of sets in \(R^{d} \). J. Comb. 1(1), 69–75 (2010). https://doi.org/10.4310/JOC.2010.v1.n1.a5

    Article  MathSciNet  MATH  Google Scholar 

  14. J.E. Wetzel, W. Wichiramala, Sectorial covers for unit arcs. Math. Mag, Math. Mag. 92(1), 42–46 (2019). https://doi.org/10.1080/0025570X.2019.1523648

    Article  MathSciNet  MATH  Google Scholar 

  15. W. Wichiramala, How support lines touch an arc. (2013) (Unpublished). https://doi.org/10.13140/RG.2.2.12818.20167

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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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Correspondence to Wacharin Wichiramala.

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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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