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}

K. Doliwka and M. Lassak, “On Relatively Short and Long Sides of Convex Pentagons,” Geometriae Dedicata, Vol. 56, No. 2, 1995, pp. 221-224.
doi:10.1007/BF01267645

M. Lassak, “On Five Points in a Plane Body Pairwise in at Least Unit Relative Distances,” Colloquia Mathematica Societatis János Bolyai, Vol. 63, North-Holland, Amsterdam, 1994, pp. 245-247.