Some notes about affine diameters of convex figures

Abstract

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.

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Soltan, V.P., Hung, N.M. Some notes about affine diameters of convex figures. Combinatorica 10, 313–317 (1990). https://doi.org/10.1007/BF02122784

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AMS subject classification (1980)

  • 52 A 30
  • 52 A 40