Obviously it is true, but I'm not sure how to prove it. I'm considering the quadrilateral inequality but so far it has not been helpful. Can anyone give me direction on how to verify $|\rho(x,z)-\rho(y,u)|\leq{\rho(x,y)+\rho(z,u)}$ for $x, y, z, u\in{X}$?
Asked
Active
Viewed 163 times
2 Answers
3
Hint: $$ \rho(x,z)\leq\rho(x,y)+\rho(y,u)+\rho(u,z)\\ \rho(y,u)\leq\rho(y,x)+\rho(x,z)+\rho(z,u) $$
Norbert
- 56,803
-
Very helpful! Thank you. – mmh0015 Jan 22 '14 at 00:43
-
@user121087, since you are new here I want to tell you that you can vote for questions and answers and even more accept answers to your questions – Norbert Jan 22 '14 at 00:44
-
Is this an obvious extension of the triangle inequality? – MathStudent1324 Apr 02 '18 at 02:13
-
1@MathStudent1324, indeed. – Norbert Apr 02 '18 at 09:35
0
We want to show:
$\rho(x,z)-\rho(y,u) \leq \rho(x,y)+\rho(z,u)$ and $-\rho(x,z)+\rho(y,u) \leq \rho(x,y)+\rho(z,u)$. Now do the "obvious" trick of "changing sides, and using the triangle inequality.
voldemort
- 13,182