On Seven Points in the Boundary of a Plane Convex Body in Large Relative Distances

By the relative distance of points p and q of a convex body C we mean the ratio of the Euclidean length of the segment pq to the half of the Euclidean length of a longest chord of C parallel to pq. We show that in the boundary of every plane convex body there exist seven points in pairwise relative distances at least 2 3 . We also give an estimate in case of three points. Finding sets of points on the sphere or ball of Euclidean n-space E such that their pairwise distances are as large as possible is a long-standing question of geometry. A generalization was presented by Lassak [6], and by Doyle, Lagarias and Randall [3]. In [3], points are considered in the boundary of the unit ball C of a Minkowski space, and their distance is measured by the Minkowski distance. In [6] we see a more general approach. Here C is allowed to be an arbitary convex body. The question is in finding configurations of points in the boundary of C which are far in the sense of the following notion of C-distance of points. For arbitrary points p, q ∈ E denote the Euclidean length of the segment pq by |pq|. Let p′q′ be a chord of C parallel to pq such that there is no longer chord of C parallel to pq. The C-distance dC(p, q) of points p and q is defined by the ratio of |pq| to 1 2 |p′q′| (see [6]). We also use the term C-length of the segment pq. If there is no doubt about C, we may use the terms relative distance of p and q, or relative length of pq. Both papers [3] and [6] show that every centrally symmetric plane convex body contains four boundary points in pairwise relative distances at least √ 2, and six boundary points whose pairwise relative distances are at least 1. Doliwka [2] proved that in the boundary of every plane convex body there exist five points in at least unit pairwise relative distances. In this paper we show a similar result about seven points in the boundary of a plane convex body. We also improve the estimate in [1] about three far boundary points. Theorem. The boundary of an arbitrary plane convex body contains seven points in pairwise relative distances at least 2 3 such that the relative distances of every two successive