Abstract
It is proved that a normed space is a Hilbert space if it possesses the property: The geometric locus of the points, for which the ratio of the distances to two given points is constant, is a sphere.
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The 2nd All-Union Symposium on Integral Geometry [in Russian], Akad. Nauk SSSR, Petrozavodsk (1967).
M. M. Day, Normed Linear Spaces, Springer-Verlag, Berlin (1962).
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Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 247–252, August, 1976.
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Danelich, I.A. Normed spaces which satisfy Apollonius' theorem. Mathematical Notes of the Academy of Sciences of the USSR 20, 696–699 (1976). https://doi.org/10.1007/BF01155877
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DOI: https://doi.org/10.1007/BF01155877