On the Relative Distances of Six Points in a Plane Convex Body

Abstract. 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. Denote the closed segment with endpoints p and q in the Euclidean plane E2 by pq, and denote the Euclidean length of pq by |pq|. Let C ⊂ E2 be a convex body. Take a chord p′q ′ of C parallel to pq such that there is no longer chord of C parallel to pq. The ratio dC(p, q) of |pq | to 12 |p′q′ | is called the C-distance of points p and q (see [5]). We also use the term C-length of the segment pq. If there is no doubt about C, we talk about the relative distance of points p and q, or the relative length of the segment pq. If C is centrally symmetric, the relative distance is the Minkowski distance in the Minkowski space whose unit ball is C. It is shown by Doyle, Lagarias and Randall [2] that among six arbitrary points of an arbitrary centrally symmetric plane convex body there is a pair in relative distance at most 1. They also showed that this value is attained for every centrally symmetric conve