The Locus And Configuration Of Points In A Relative Metric Space

Let C C En be a plane convex body, for arbitrary points a,b∈En, denote by|ab|the Euclidean length of the line-segment ab. Let a1b1be a longest chord of C parallel to the line-segment ab, denote by a1b1the affine diameter of C parallel to ab. The C-distance dC(a, b) between the points a and b is defined by the ratio of|ab|to1/2|a1b1|. If there is no confusion about C, we may use the terms relative distance between a and b. Langi conjectured that there exists no plane convex body whose boundary contains nine points at pairwise relative distance greater than4sin π/18, and that there exists no plane convex body whose boundary contains ten (or eleven) points at pairwise relative distances greater than|.We know that an ellipse is the locus of all points in the plane the sum of whose Euclidean distance from two fixed points is a given positive constant. First, in this paper we define an ellipse in a relative metric space by putting "relative distance" in the part of "Euclidean distance" and prove the following theorem.Theorem1Let T=abc be a triangle, and let f1,f2be two fixed points with f1f2‖ab. Then for any real number Τ>|f1f2|, the locus of the points p satisfying the condition,dΤ(p, f1)+dΤ(p, f2)=Τ, is either a centrally symmetric decagon (when Τ≠2dΤ(f1, f2)) or a centrally symmetric octagon (when r=2dΤ(f1,f2)).Second, we construct a nonagon whose relative distances of arbitrary two consecutive ver-tices are equal to (?)-1, which improves the bound about Φ9(C) and construct a decagon whose relative distances of arbitrary two consecutive vertices are greater than|, which disproves the conjecture about Φ10(C),we also through the strict logical reasoning prove the conjecture about Φ11(C). That is to say, we prove the following theorems.Theorem2There exists a convex nonagon N with the property that the relative distance of any two its vertices equals (?)-1. That is, Φ9(C)≥(?)-1.Theorem3There exists a convex decagon D0with the property that the relative distance of any two its vertices is greater than|. That is, Φ10(C)>2/3.Theorem4Every convex hendecagon has two consecutive vertices with relative distance at most|. That is, Φ11(C)=2/3.