The Triangle Inequality and Its Applications in the Relative Metric Space

Affiliation(s)

College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, China.

Department of Mathematics, Texas State University-San Marcos Texas State, San Marcos, USA.

College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, China.

Department of Mathematics, Texas State University-San Marcos Texas State, San Marcos, USA.

ABSTRACT

Let *C* be a plane convex body. For arbitrary
points ,* a,b ∈E*^{ n}denote by │*ab*│ the Euclidean length of the line-segment ab. Let *a*_{1}*b*_{1} 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 *a*_{1} and *b*_{1}. In this note we prove the
triangle inequality in *E*^{2} 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.

Cite this paper

Z. Su, S. Li and J. Shen, "The Triangle Inequality and Its Applications in the Relative Metric Space,"*Open Journal of Discrete Mathematics*, Vol. 3 No. 3, 2013, pp. 127-129. doi: 10.4236/ojdm.2013.33023.

Z. Su, S. Li and J. Shen, "The Triangle Inequality and Its Applications in the Relative Metric Space,"

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.

[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.