TITLE:
The Triangle Inequality and Its Applications in the Relative Metric Space
AUTHORS:
Zhanjun Su, Sipeng Li, Jian Shen
KEYWORDS:
Relative Distance; Triangle Inequality; Hexagon
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.3 No.3,
July
12,
2013
ABSTRACT: 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 dc(a,b) between the points a and b is the ratio of the Euclidean distance between aand bto the half of the Euclidean distance between a1 and b1. In this note we prove the
triangle inequality in E2 with the relative metric dc( .,.), 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 dp( .,.). In addition, we prove that
every convex hexagon has two pairs of consecutive vertices with relative distances
at least 1.