The Triangle Inequality and Its Applications in the Relative Metric Space

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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.

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.