Metric and Norms: Finding a Violation of the Triangle Inequality

[SimbaTheLion]

Homework Statement

The Wikipedia part of question 5 here:

http://www.dpmms.cam.ac.uk/site2002/Teaching/IB/MetricTopologicalSpaces/2007-2008/Examples1.pdf

Homework Equations

All relevant information is given in the question above.

The Attempt at a Solution

I'm trying to simplify the problem by considering only 2-dimensional vectors rather than n-dimensional ones. But I've considered dozens of combinations of vectors, norms and metrics, none of which are working. I think that what Wikipedia says is correct in that d-tilde(x, y) = 0 <=> x = y, and d-tilde(x, y) = d-tilde(y, x), so the problem lies in the triangle inequality being violated I presume. But after two days of trying so many things out (and killing lots of trees for paper :P ) I still haven't come up with a single counter-example.

Can someone please tell me which vector/norm/metric combination I should be considering?

Thanks a lot!

 

[Office_Shredder]

My rule of thumb. If wikipedia says it's a metric, it probably is. Have you considered proving it doesn't violate the triangle inequality? Usually if you try and fail it at least gives you an idea of why it doesn't satisfy the triangle inequality (based on where you get stuck)

 

[Dick]

My rule of thumb. If wikipedia says it's a metric, it probably is. Have you considered proving it doesn't violate the triangle inequality? Usually if you try and fail it at least gives you an idea of why it doesn't satisfy the triangle inequality (based on where you get stuck)

According to the footnote, wikipedia has retracted the statement. Now what do you say? Just a warning.

 

[Office_Shredder]

According to the footnote, wikipedia has retracted the statement. Now what do you say? Just a warning.

I would still stand by my statement. If you can't come up with a counterexample, try to start a proof to demonstrate it's true and see where you get stuck. If the statement is false you usually get stuck at the point where the statement ends up being false, so you can see what kind of condition is necessary for the metric space you want to look at

 

[Billy Bob]

If you can't come up with a counterexample, try to start a proof to demonstrate it's true and see where you get stuck.

I tried this, and I think the metrics won't make any difference. I think the norm has to be unusual. I have an idea, but I haven't finished.