Skip to main content
SpringerLink
Log in

Stability of the Simplex Bound for Packings by Equal Spherical Caps Determined by Simplicial Regular Polytopes

  • Conference paper
  • First Online:
Discrete Geometry and Symmetry (GSC 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 234))

Included in the following conference series:

  • 628 Accesses

Abstract

It is well known that the vertices of any simplicial regular polytope in \(\mathbb R^d\) determine an optimal packing of equal spherical balls in \(S^{d-1}\). We prove a stability version of optimal order of this result.

Research is supported in parts by NKFIH grants 109789, 121649 and 116451.

Research is partially supported by NSF grant DMS-1400876

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J. Aczél, Solution to problem 35, I. (Hungarian). Mat. Lapok, 3, 94-95 (1952)

    Google Scholar 

  2. N.N. Andreev, A spherical code. Russ. Math. Surv. 54, 251–253 (1999)

    Article  MathSciNet  Google Scholar 

  3. K. Bezdek, Classical Topics in Discrete Geometry (Springer, 2010)

    Book  Google Scholar 

  4. K. Böröczky, Packing of spheres in spaces of constant curvature. Acta Math. Hung. 32, 243–261 (1978)

    Article  MathSciNet  Google Scholar 

  5. K. Böröczky Jr., Finite Packing and Covering (Cambridge Unversity Press, 2004)

    Google Scholar 

  6. P. Boyvalenkov, D. Danev, Uniqueness of the 120-point spherical 11-design in four dimensions. Arch. Math. 77, 360–268 (2001)

    Article  MathSciNet  Google Scholar 

  7. S.A. Chepanov, S.S. Ryshkov, N.N. Yakovlev, On the disjointness of point systems. (Russian) Trudy Mat. Inst. Steklov. 196, 147–155 (1991)

    MathSciNet  MATH  Google Scholar 

  8. J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups (Springer, Berlin, New York, 1998)

    MATH  Google Scholar 

  9. H.S.M. Coxeter, Regular Polytopes (Originally Published in 1947) (Dover, 1973)

    Google Scholar 

  10. H. Davenport, Gy. Hajós, Problem 35 (Hungarian). Mat. Lapok, 2, 68 (1951)

    Google Scholar 

  11. T. Ericson, V. Zinoviev, Codes on Euclidean Spheres (North-Holland, 2001)

    Google Scholar 

  12. G. Fejes Tóth, Packing and Covering, in CRC Handbook on Discrete and Computational Geometry, ed. by E. J. Goodman, J. O’Rourke (CRC Press, 2004)

    Google Scholar 

  13. L. Fejes Tóth, On the densest packing of spherical caps. Am. Math. Monthly 56, 330–331 (1949)

    Article  MathSciNet  Google Scholar 

  14. L. Fejes Tóth, On the volume of a polyhedron in non-Euclidean spaces. Publ. Math. Debrecen 4, 256–261 (1956)

    MathSciNet  MATH  Google Scholar 

  15. L. Fejes Tóth, Regular Figures (Pergamon Press, Oxford, 1964)

    Google Scholar 

  16. L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum (Springer, Berlin, 1972)

    Google Scholar 

  17. H.W.E. Jung, Über die kleineste Kugel die eine räumliche Figur einschliesst. J. Reine ang. Math. 123, 241–257 (1901)

    MATH  Google Scholar 

  18. P. McMullen, E. Schulte, Abstract Regular Polytopes (Cambridge University Press. 2002)

    Google Scholar 

  19. O. Musin, The kissing number in four dimensions. Ann. Math. 168, 1–32 (2008)

    Article  MathSciNet  Google Scholar 

  20. R.A. Rankin, The closest packing of spherical caps in \(n\) dimensions. Proc. Glasgow Math. Assoc. 2, 139–144 (1955)

    Article  MathSciNet  Google Scholar 

  21. C.A. Rogers, Packing and Covering (Cambridge University Press, Cambridge, 1964)

    MATH  Google Scholar 

  22. T. Szele, Solution to Problem 35, II. (Hungarian). Mat. Lapok, 3, 95 (1952)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Roland Eötvös University, Pázmány Péter sétány 1/C, H-1117, Budapest, Hungary

    Károly Böröczky & Ágnes Kovács

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, H-1053, Budapest, Hungary

    Károly J. Böröczky

  3. Central European University, Nádor u. 9, H-1051, Budapest, Hungary

    Károly J. Böröczky

  4. School of Mathematical & Statistical Sciences, The University of Texas Rio Grande Valley, One West University Blvd, Brownsville, TX, USA

    Alexey Glazyrin

Authors
  1. Károly Böröczky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Károly J. Böröczky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexey Glazyrin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ágnes Kovács
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Károly J. Böröczky .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Auckland, Auckland, New Zealand

    Marston D. E. Conder

  2. Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada

    Antoine Deza

  3. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

    Asia Ivić Weiss

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Böröczky, K., Böröczky, K.J., Glazyrin, A., Kovács, Á. (2018). Stability of the Simplex Bound for Packings by Equal Spherical Caps Determined by Simplicial Regular Polytopes. In: Conder, M., Deza, A., Weiss, A. (eds) Discrete Geometry and Symmetry. GSC 2015. Springer Proceedings in Mathematics & Statistics, vol 234. Springer, Cham. https://doi.org/10.1007/978-3-319-78434-2_2

Download citation

Publish with us

Policies and ethics

Search

Navigation