Geometric and Functional Analysis

, Volume 27, Issue 6, pp 1367–1377 | Cite as

Proof of László Fejes Tóth’s zone conjecture

  • Zilin JiangEmail author
  • Alexandr Polyanskii


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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  1. AHKM02.
    Aharoni R., Holzman R., Krivelevich M., Meshulam R.: Fractional planks. Discrete Comput. Geom., 27(4), 585–602 (2002)MathSciNetCrossRefzbMATHGoogle 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.Google Scholar
  6. Bal01.
    Ball K.M.: The complex plank problem. Bull. London Math. Soc., 33(4), 433–442 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Ban50.
    Bang T.: On covering by parallel-strips. Mat. Tidsskr. B., 1950, 49–53 (1950)MathSciNetzbMATHGoogle Scholar
  8. Ban51.
    Bang. T.: A solution of the “plank problem”. Proc. Amer. Math. Soc., 2, 990–993 (1951)MathSciNetzbMATHGoogle 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).Google Scholar
  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).Google Scholar
  12. Bog61.
    N. Bognár. On W. Fenchel’s solution of the plank problem. Acta Math. Acad. Sci. Hungar., 12 (1961), 269–270.Google Scholar
  13. BS10.
    Bezdek K., Schneider R.: Covering large balls with convex sets in spherical space. Beiträge Algebra Geom., 51(1), 229–235 (2010)MathSciNetzbMATHGoogle Scholar
  14. FVZ16.
    Fodor F., Vígh V., Zarnócz T: Covering the sphere by equal zones. Acta Math. Hungar., 149(2), 478–489 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. GG45.
    Goodman A.W., Goodman R.E.: A circle covering theorem. Amer. Math. Monthly, 52, 494–498 (1945)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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.Google Scholar
  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.Google Scholar
  20. Ros72.
    Rosta V.: An extremal distribution of three great circles. Mat. Lapok, 23, 161–162 (1972)MathSciNetGoogle Scholar
  21. Tar32.
    A. Tarski. Uwagi o stopniu równowa zności wielokątów [Remarks on the degree of equivalence of polygons]. Parametr, 2 (1932), 310–314.Google Scholar
  22. Tót73.
    Tóth L.F.: Research problems: Exploring a planet. Amer. Math. Monthly, 80(9), 1043–1044 (1973)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsTechnion – Israel Institute of TechnologyTechnion CityIsrael
  2. 2.Moscow Institute of Physics and Technology and Institute for Information Transmission Problems RASMoscowRussia

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