The Triangle Inequality and Its Applications in the Relative Metric Space

Abstract

Let C be a plane convex body. For arbitrary points , a,b ∈E n denote by │ ab │ the Euclidean length of the line-segment ab. Let a 1 b 1 be a longest chord of C parallel to the line-segment ab . The relative distance d c (a,b) between the points a and b is the ratio of the Euclidean distance between a and b to the half of the Euclidean distance between a 1 and b 1 . In this note we prove the triangle inequality in E 2 with the relative metric d c( . , . ) , and apply this inequality to show that 6≤l(P)≤8 , where l(P) is the perimeter of the convex polygon P measured in the metric d p ( . , . ) . In addition, we prove that every convex hexagon has two pairs of consecutive vertices with relative distances at least 1.