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.
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(i)
γ(x)≥2 for every pointx ∈ intM.
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(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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References
B.Grünbaum: Measures of symmetry for convex sets, In V. Klee ed., Proceedings Symp. Pure Math., VolVII.,Amer. Math. Soc., (1963), 233–270.
B. Grünbaum: Continuous families of curves,Canad. J. Math.,18 (1966) 529–537.
T. Zamfirescu: On continuous families of curves.VI. Geom. Dedic.,10 (1981) 205–217.
P. C. Hammer: Diameters of convex bodies,Proc. Amer. Math. Soc.,5 (1954) 304–306.
H. G. Eggleston: Some properties of triangles as external convex curves,J. London Math. Soc.,28 (1953) 32–36.