Periodica Mathematica Hungarica

, Volume 39, Issue 1–3, pp 7–15 | Cite as


  • Károly Bezdek
  • Grigoriy Blekherman


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.


Main Tool Strong Version Spherical Geometry Convex Polyhedron Adjacent Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. Bateman AND P. ErdŐs, Geometrical extrema suggested by a lemma of Besicovitch, Amer. Math. Monthly 58no. 5 (1951), 306–314.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    A. Bezdek AND F. Fodor, Minimal diameter of certain sets in the plane, J. Comb. Theory, Ser. A 85no. 1 (1999), 105–111.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    K. Bezdek, G. Blekherman, R. Connelly AND B. CsikÓs, The polyhedral Tammes problem, Arch. Math. (to appear), 1–8.Google Scholar
  4. [4]
    H. T. Croft, On 6–points configuration in 3–space, J. London Math. Soc. 36 (1961), 289–306.zbMATHMathSciNetGoogle Scholar
  5. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    P. ErdŐs, Some unsolved problems, Michigan Math. J. 4 (1957), 291–300.MathSciNetCrossRefGoogle Scholar
  7. [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. [8]
    G. A. Kabatiansky AND V. I. Levenshtein, Bounds for packings on a sphere and in space, Problemy Peredachi Informatsii 14 (1978), 3–25.MathSciNetGoogle Scholar
  9. [9]
    V. L. Klee, Unsolved problems in intuitive geometry, Hectographical lectures, Seattle, 1960.Google Scholar
  10. [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.Google Scholar
  11. [11]
    J. Pach, personal communication.Google Scholar
  12. [12]
    J. PÁl, Ein minimumproblem für Ovale, Math. Ann. 83 (1921), 311–319.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    K. SchÜtte, Minimale Durchmesser endlicher Punktmengen mit vorgeschriebenen Mindenstabstand, Math. Ann. 150 (1963), 91–98.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    R. Webster, Convexity, Oxford University Press, 1994.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Károly Bezdek
    • 1
  • Grigoriy Blekherman
    • 1
  1. 1.Department of GeometryEötvös UniversityHungary E-mail

Personalised recommendations