Metric and Norms: Finding a Violation of the Triangle Inequality

In summary, the conversation discusses a problem involving metric spaces and the violation of the triangle inequality. The original question asks for a specific vector/norm/metric combination to consider. The participants suggest trying to prove the statement or finding a counterexample, and also mention that Wikipedia has retracted the statement. They also suggest that the norm may be the key factor in this problem.
  • #1
SimbaTheLion
8
0

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!
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
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)
 
  • #3
Office_Shredder said:
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.
 
  • #4
Dick said:
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
 
  • #5
Office_Shredder said:
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.
 

FAQ: Metric and Norms: Finding a Violation of the Triangle Inequality

What is a metric space?

A metric space is a mathematical concept used to define distances between objects. It consists of a set of objects and a distance function that assigns a non-negative real number to every pair of objects in the set. This distance function must satisfy certain properties, such as the distance between two objects being 0 only if the two objects are the same, and the distance between two objects being symmetric (i.e. the distance from object A to object B is the same as the distance from object B to object A).

What is a norm in a metric space?

A norm is a function that assigns a non-negative real number to an object in a metric space. This function measures the length or size of the object. It must also satisfy certain properties, such as being 0 only when the object is the zero vector, and satisfying the triangle inequality (i.e. the length of the sum of two objects is less than or equal to the sum of their individual lengths).

How are metric spaces and norms related?

In a metric space, the distance between two objects can be measured using a norm. This is because a norm is a function that satisfies the properties required for a distance function in a metric space. In other words, a norm provides a way to measure the distance between two objects in a metric space.

What are some common examples of metric spaces and norms?

Some common examples of metric spaces include Euclidean space, which is used to measure distances in three-dimensional space, and the space of real numbers, which is used to measure distances on a number line. Examples of norms include the Euclidean norm, which measures the length of a vector in Euclidean space, and the absolute value function, which measures the distance of a number from 0 on the number line.

Why are metric spaces and norms important in mathematics and science?

Metric spaces and norms provide a framework for measuring distances and sizes in mathematical and scientific contexts. They are used in various fields such as geometry, physics, and statistics to study and analyze objects and their properties. They also have practical applications in areas such as data analysis, optimization, and machine learning.

Similar threads

Understanding of the Metric Space axioms - (axiom 2 only)
Replies
19
Views
2K
Metric space of continuous & bounded functions is complete?
Replies
9
Views
2K
Complex numbers sequences/C is a metric space
Replies
15
Views
2K
I Shouldn't this definition of a metric include a square root?
Replies
8
Views
2K
Is This Approach Valid for Proving the Discrete Metric in a Metric Space?
Replies
2
Views
1K
Finding the closure of some metric spaces
Replies
7
Views
1K
Inner Product, Triangle and Cauchy Schwarz Inequalities
Replies
5
Views
3K
Show the following is a metric
Replies
5
Views
1K
Is the composition of a function and a metric a metric?
Replies
1
Views
2K
Metric Spaces, Triangle Inequality
Replies
1
Views
3K

Hot Threads

Recent Insights

Back
Top