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.
Alonso, J. and Benítez, C., Orthogonality in normed linear spaces: a survey. Part I: main properties, Extracta Math., 3 (1988), no. 1, 1–15. MR91e:46021a
Benítez C., 'Orthogonality in normed linear spaces: a survey. Part I: main properties' (1988) 3Extracta Math.: 1-15.
Benítez C.Orthogonality in normed linear spaces: a survey. Part I: main propertiesExtracta Math.19883115)| false