Let C be a convex body in the Euclidean plane. By the relative distance of points p and q we mean the ratio of the Euclidean distance of p and q to the half of the Euclidean length of a longest chord of C parallel to pq. In this note we find the least upper bound of the minimum pairwise relative distance of six points in a plane convex body.
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.