Skip to main content
SpringerLink
Log in

Proof of László Fejes Tóth’s zone conjecture

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Technion – Israel Institute of Technology, Technion City, Haifa, 32000, Israel

    Zilin Jiang

  2. Moscow Institute of Physics and Technology and Institute for Information Transmission Problems RAS, Moscow, Russia

    Alexandr Polyanskii

Authors
  1. Zilin Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexandr Polyanskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-017-0427-6

Search

Navigation