OJDMOpen Journal of Discrete Mathematics2161-7635Scientific Research Publishing10.4236/ojdm.2013.33023OJDM-34501ArticlesPhysics&Mathematics The Triangle Inequality and Its Applications in the Relative Metric Space hanjunSu1*SipengLi1*JianShen2*College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, ChinaDepartment of Mathematics, Texas State University-San Marcos Texas State, San Marcos, USA* E-mail:suzj888@163.com(HS);sipengli@126.com(SL);js48@txstate.edu(JS);020720130303127129January 10, 2013April 20, 2013 May 16, 2013© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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 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 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.

Relative Distance; Triangle Inequality; Hexagon
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