%0 Journal Article %T The Triangle Inequality and Its Applications in the Relative Metric Space %A Zhanjun Su %A Sipeng Li %A Jian Shen %J Open Journal of Discrete Mathematics %P 127-129 %@ 2161-7643 %D 2013 %I Scientific Research Publishing %R 10.4236/ojdm.2013.33023 %X

Let C be a plane convex body. For arbitrary points , a,b ﹋Eⁿdenote by 岫ab岫 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