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  • Author or Editor: Senlin Wu x
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We give a geometric characterization of inner product spaces among all finite dimensional real Banach spaces via concurrent chords of their spheres. Namely, let x be an arbitrary interior point of a ball of a finite dimensional normed linear space X. If the locus of the midpoints of all chords of that ball passing through x is a homothetical copy of the unit sphere of X, then the space X is Euclidean. Two further characterizations of the Euclidean case are given by considering parallel chords of 2-sections through the midpoints of balls.

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Abstract  

Studying the relation between the length of a chord of the unit circle and the length of the arc corresponding to it, some new characterizations of the Euclidean plane among all normed planes are obtained. All these results yield characterizations of inner product spaces in higher dimensions.

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