On the relative distances of seven points in a plane convex body

Let C be a convex body in the Euclidean plane. The relative distance of points p and q is twice the Euclidean distance of p and q divided by the Euclidean length of a longest chord in C with the direction, say, from p to q. We prove that, among any seven points of a plane convex body, there are two points at relative distance at most one, and one cannot be replaced by a smaller value. We apply our result to determine the diameter of point sets in normed planes. Zsolt Lángi: Partially supported by the Hung. Nat. Sci. Found. (OTKA), grant no. T043556 and T037752 and by the Alberta Ingenuity Fund.