Skip to main content
SpringerLink
Log in

DANZER-GRüNBAUM'S THEOREM REVISITED

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

In this paper we prove some stronger versions of Danzer-Grünbaum's theorem including the following stability-type result. For 0 < α < 14π/27 the maximum number of vertices of a convex polyhedron in E 3 such that all angles between adjacent edges are bounded from above by α is 8. One of the main tools is the spherical geometry version of Pál's theorem.

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. P. Bateman AND P. ErdŐs, Geometrical extrema suggested by a lemma of Besicovitch, Amer. Math. Monthly 58no. 5 (1951), 306–314.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Bezdek AND F. Fodor, Minimal diameter of certain sets in the plane, J. Comb. Theory, Ser. A 85no. 1 (1999), 105–111.

    Article  MATH  MathSciNet  Google Scholar 

  3. K. Bezdek, G. Blekherman, R. Connelly AND B. CsikÓs, The polyhedral Tammes problem, Arch. Math. (to appear), 1–8.

  4. H. T. Croft, On 6–points configuration in 3–space, J. London Math. Soc. 36 (1961), 289–306.

    MATH  MathSciNet  Google Scholar 

  5. L. Danzer AND B. GrÜnbaum, Über zwei Probleme bezüglich konvexer Körper von P. Erdős und V. L. Klee, Math. Z. 79 (1962), 95–99.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. ErdŐs, Some unsolved problems, Michigan Math. J. 4 (1957), 291–300.

    Article  MathSciNet  Google Scholar 

  7. P. ErdŐs AND Z. FÜredi, The greatest angle among n points in the d-dimensional Euclidean space, Combinatorial Math. North-Holland Math. Stud., vol. 75, North-Holland, Amsterdam and New York, 1983.

    Google Scholar 

  8. G. A. Kabatiansky AND V. I. Levenshtein, Bounds for packings on a sphere and in space, Problemy Peredachi Informatsii 14 (1978), 3–25.

    MathSciNet  Google Scholar 

  9. V. L. Klee, Unsolved problems in intuitive geometry, Hectographical lectures, Seattle, 1960.

  10. E. Makai Jr. AND H. Martini, On the number of antipodal or strictly antipodal pairs of points infinite subsets of R d, DIMACS Ser. in Discrete Math. and Theoretical Comp. Sci. vol. 4, 1991.

  11. J. Pach, personal communication.

  12. J. PÁl, Ein minimumproblem für Ovale, Math. Ann. 83 (1921), 311–319.

    Article  MATH  MathSciNet  Google Scholar 

  13. K. SchÜtte, Minimale Durchmesser endlicher Punktmengen mit vorgeschriebenen Mindenstabstand, Math. Ann. 150 (1963), 91–98.

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Webster, Convexity, Oxford University Press, 1994.

Download references

Author information

Authors and Affiliations

  1. Department of Geometry, Eötvös University, H-1053, Budapest Kecskeméti utca 10-12, Hungary E-mail

    Károly Bezdek & Grigoriy Blekherman

Authors
  1. Károly Bezdek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Grigoriy Blekherman
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bezdek, K., Blekherman, G. DANZER-GRüNBAUM'S THEOREM REVISITED. Periodica Mathematica Hungarica 39, 7–15 (2000). https://doi.org/10.1023/A:1004878520642

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1004878520642

Keywords

Search

Navigation