The Triangle Inequality and Its Applications in the Relative Metric Space

Zhanjun Su, Sipeng Li, Jian Shen

College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, China.
Department of Mathematics, Texas State University-San Marcos Texas State, San Marcos, USA.

DOI: 10.4236/ojdm.2013.33023                

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 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.

Keywords

Relative Distance; Triangle Inequality; Hexagon

Conflicts of Interest

The authors declare no conflicts of interest.

References

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