Proof of László Fejes Tóth’s zone conjecture

Abstract

A zone of width ω on the unit sphere is the set of points within spherical distance ω/2 of a given great circle. We show that the total width of any collection of zones covering the unit sphere is at least π, answering a question of Fejes Tóth from 1973.

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References

  1. AHKM02

    Aharoni R., Holzman R., Krivelevich M., Meshulam R.: Fractional planks. Discrete Comput. Geom., 27(4), 585–602 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  2. AK12

    A. Akopyan and R. Karasev. Kadets-type theorems for partitions of a convex body. Discrete Comput. Geom., (3)48 (2012), 766–776. arXiv:1106.5635 [math.CO].

  3. AKP14

    A. Akopyan, R. Karasev, and F. Petrov. Bang’s Problem and Symplectic Invariants (2014). arXiv:1404.0871 [math.MG].

  4. Bal91

    K. Ball. The plank problem for symmetric bodies. Invent. Math., (3)104 (1991), 535–543. arXiv:math/9201218 [math.MG].

  5. Bal92

    K. Ball. A lower bound for the optimal density of lattice packings. Internat. Math. Res. Notices, (10) (1992), 217–221.

  6. Bal01

    Ball K.M.: The complex plank problem. Bull. London Math. Soc., 33(4), 433–442 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  7. Ban50

    Bang T.: On covering by parallel-strips. Mat. Tidsskr. B., 1950, 49–53 (1950)

    MathSciNet  MATH  Google Scholar 

  8. Ban51

    Bang. T.: A solution of the “plank problem”. Proc. Amer. Math. Soc., 2, 990–993 (1951)

    MathSciNet  MATH  Google Scholar 

  9. Bez03

    A. Bezdek. Covering an annulus by strips. Discrete Comput. Geom., (2)30 (2003), 177–180. U.S.-Hungarian Workshops on Discrete Geometry and Convexity (Budapest, 1999/Auburn, AL, 2000).

  10. Bez13

    K. Bezdek. Tarski’s plank problem revisited. In: Geometry—Intuitive, Discrete, and Convex, Vol. 24 of Bolyai Soc. Math. Stud., pP. 45–64. János Bolyai Math. Soc., Budapest, 2013. arXiv:0903.4637 [math.MG].

  11. BMP05

    P. Brass, W. Moser, and J. Pach. Research Problems in Discrete Geometry. Springer, New York (2005).

  12. Bog61

    N. Bognár. On W. Fenchel’s solution of the plank problem. Acta Math. Acad. Sci. Hungar., 12 (1961), 269–270.

  13. BS10

    Bezdek K., Schneider R.: Covering large balls with convex sets in spherical space. Beiträge Algebra Geom., 51(1), 229–235 (2010)

    MathSciNet  MATH  Google Scholar 

  14. FVZ16

    Fodor F., Vígh V., Zarnócz T: Covering the sphere by equal zones. Acta Math. Hungar., 149(2), 478–489 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  15. GG45

    Goodman A.W., Goodman R.E.: A circle covering theorem. Amer. Math. Monthly, 52, 494–498 (1945)

    MathSciNet  Article  MATH  Google Scholar 

  16. Kad05

    V. Kadets. Coverings by convex bodies and inscribed balls. Proc. Amer. Math. Soc., (5)133 (2005), 1491–1495. arXiv:math/0312133 [math.FA].

  17. Lin74

    Linhart J.: Eine extremale Verteilung von Grosskreisen. Elem. Math., 29, 57–59 (1974)

    MathSciNet  MATH  Google Scholar 

  18. MMS14

    A. McFarland, J. McFarland, and J.T. Smith, editors. Alfred Tarski. Early work in Poland—geometry and teaching. Birkhäuser/Springer, New York (2014). With a bibliographic supplement, Foreword by Ivor Grattan-Guinness.

  19. Moe32

    H. Moese. Przyczynek do problemu A. Tarskiego: “O stopniu równowaonosci wielokątów” [A contribution to the problem of A. Tarski, “On the degree of equivalence of polygons”]. Parametr, 2 (1932), 305–309.

  20. Ros72

    Rosta V.: An extremal distribution of three great circles. Mat. Lapok, 23, 161–162 (1972)

    MathSciNet  Google Scholar 

  21. Tar32

    A. Tarski. Uwagi o stopniu równowa zności wielokątów [Remarks on the degree of equivalence of polygons]. Parametr, 2 (1932), 310–314.

  22. Tót73

    Tóth L.F.: Research problems: Exploring a planet. Amer. Math. Monthly, 80(9), 1043–1044 (1973)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Zilin Jiang.

Additional information

Z. Jiang was supported in part by ISF Grant Nos. 1162/15, 936/16.

A. Polyanskii was supported in part by ISF Grant No. 409/16, and by the Russian Foundation for Basic Research through Grant Nos. 15-01-99563 A, 15-01-03530 A. The work was done when A. Polyanskii was a postdoctoral fellow at the Technion.

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Jiang, Z., Polyanskii, A. Proof of László Fejes Tóth’s zone conjecture. Geom. Funct. Anal. 27, 1367–1377 (2017). https://doi.org/10.1007/s00039-017-0427-6

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