Some notes about affine diameters of convex figures


Given a pointx in a convex figureM, letγ(x) denote the number of all affine diameters ofM passing throughx. It is shown that, for a convex figureM, the following conditions are equivalent.

  1. (i)

    γ(x)≥2 for every pointx ∈ intM.

  2. (ii)

    eitherγ(x)≡3 orγ(x)≡∞ on intM. Furthermore, the setB={x ∈ intM:γ(x) is either odd or infinite } is dense inM.

This is a preview of subscription content, access via your institution.


  1. [1]

    B.Grünbaum: Measures of symmetry for convex sets, In V. Klee ed., Proceedings Symp. Pure Math., VolVII.,Amer. Math. Soc., (1963), 233–270.

  2. [2]

    B. Grünbaum: Continuous families of curves,Canad. J. Math.,18 (1966) 529–537.

    Google Scholar 

  3. [3]

    T. Zamfirescu: On continuous families of curves.VI. Geom. Dedic.,10 (1981) 205–217.

    Google Scholar 

  4. [4]

    P. C. Hammer: Diameters of convex bodies,Proc. Amer. Math. Soc.,5 (1954) 304–306.

    Google Scholar 

  5. [5]

    H. G. Eggleston: Some properties of triangles as external convex curves,J. London Math. Soc.,28 (1953) 32–36.

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Soltan, V.P., Hung, N.M. Some notes about affine diameters of convex figures. Combinatorica 10, 313–317 (1990).

Download citation

AMS subject classification (1980)

  • 52 A 30
  • 52 A 40