On a conjecture about nine points in the boundary of a plane convex body at pairwise relative distances not greater than $${4 \, {\rm sin}\frac{\pi}{18}}$$

Abstract

The relative distance of points a and b (or the relative length of the line-segment ab) in a convex body C is the ratio of the length of the line-segment ab to the half of the length of a longest chord of C parallel to ab. Langi conjectured that there exists no plane convex body whose boundary contains nine points at pairwise relative distances greater than \({4\sin\frac{\pi}{18} = 0.69459\ldots}\). In this paper we disprove this conjecture. Moreover, we find an infinite sequence of positive integers n for which there is a convex n-gon with no relatively long sides.