All Title Author
Keywords Abstract


The Triangle Inequality and Its Applications in the Relative Metric Space

DOI: 10.4236/ojdm.2013.33023, PP. 127-129

Keywords: Relative Distance, Triangle Inequality, Hexagon

Full-Text   Cite this paper   Add to My Lib

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 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

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.

Full-Text

comments powered by Disqus