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

NOTES

Shen’s research was partially supported by NSF (CNS 0835834, DMS 1005206) and Texas Higher Education Coordinating Board (ARP 003615-0039-2007).

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	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
[2]	I. Fáry and E. Makai Jr., “Isoperimetry in Variable Metric,” Studia Scientiarum Mathematicarum Hungarica, Vol. 17, 1982, pp. 143-158.
[3]	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.