Let C be a plane convex body. For arbitrary points , a,b ∈E ⁿdenote by │ab│ the Euclidean length of the line-segment ab. Let a₁b₁ 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 a₁ and b₁. In this note we prove the triangle inequality in E² 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.