On the relative distances of eleven points in the boundary of a plane convex body

Let C be a plane convex body. The relative distance (or C -distance) of points a , b ∈ R 2 is defined by the ratio of the Euclidean length of the line-segment a b to half of the Euclidean length of a longest chord of C , parallel to a b . It was conjectured that there exists no plane convex body whose boundary contains eleven points at pairwise relative distances greater than 2 3 . We give an affirmative answer to this conjecture.