Talk:Triangle inequality

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Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 20:32, 20 May 2007 (UTC)

What about |x+y| <= |x|+|y|, on the real line. I've seen that called the triangle inequality. Is it just by analogy to Euclidean space, and from length to absolute value?

For me this is just a poor formulation. More direct is |x|-|y| <= |x+y|<=|x|+|y|, this is the actual "triangle."

Contents 1 Consequences unclear and Seem Contradictory 2 Intuition 3 Proof 4 Article Deletion 5 Contrary Definition

[edit] Consequences unclear and Seem Contradictory

On the inverse triangle inequality, it is given | |x| - |y| | <= |x - y| This makes sense to me. But in Consequences, some technical specifications are given and then a seemingly contradictory statement is given. Either the format of this section has gone awry, or something else needs to be clarified. Someone more knowledgable, please? 128.171.31.11 (talk) 09:13, 18 February 2008 (UTC)

There is no contradiction. I'm not sure how you came to that conclusion - all the statements given in the article are true, although some use sums and some use differences, and some use norms while others use distance functions. In particular, both "| |x| - |y| | <= |x - y|" and "| |x| - |y| | <= |x + y|" are true (in fact | |x| - |y| | is at most the smallest of these two quantities). Dcoetzee 01:33, 21 February 2008 (UTC)

[edit] Intuition

I think that the triangle inequality is also quite intuitive, because it is impossible to draw a triangle if the "base" is longer then the sum of the two other stems. Given that my English is not good enough I am afraid to formulate this is a proper way, but it would be good to add such a section to the article. —The preceding unsigned comment was added by 89.80.147.73 (talk) 10:16, 21 January 2007 (UTC).

It is intuitive in R³, and other similar vector spaces. However, the result is more general and can be applied in any well defined inner product space - it can be shown from Cauchy Schwarz, and as such applies to many more situations that just a simple triangle in a Euclidean Geometry. It can easily be extended to more abstract versions of the inner product - for example it can be used in quantum mechanics when considering "overlap integrals". The Young Ones (talk) 20:48, 25 April 2008 (UTC)

[edit] Proof

You can not state the triangle inequality without giving a proof. The article is severly lacking and should be deleted. —Preceding unsigned comment added by 65.32.93.17 (talk • contribs) 23:14, 30 July 2007

I wouldn't say it needs to be deleted, but would suggest it is in need of a proof. The Young Ones (talk) 20:48, 25 April 2008 (UTC)

Have expanded this to include a short 'proof' using Cauchy-Schwarz Inequality. Let me know what you think. The Young Ones (talk) 20:59, 25 April 2008 (UTC)

[edit] Article Deletion

I nominate this article for deletion on the basis that it is extremely incomplete and makes unfounded statements without proof. --anon

There is no need to have proof in an encyclopedia. This is not a book. Oleg Alexandrov (talk) 03:00, 13 August 2007 (UTC)

Whilst Wikipedia is not a Maths text book, I don't think it is at all unreasonable to request to have a proof in an article about a theorem, especially if the proof is very short as in this case. From a Methematical perspective, the proof is the most important part of the theorem, and, if at all reasonable, as an Encyclopedia we have a duty to document it. So many thanks to User:The Young Ones for adding a proof in yesterday! -- simxp (talk) 14:37, 26 April 2008 (UTC)

[edit] Contrary Definition

The definition given seems to imply the exclusion of colinear vectors and points.

My reading, including the reading of articles in refereed mathematical and physics journals include mathematical analysies that depend on colinear vectors and points being accepted by the triangle inequality theorem.

Riley K. F., Hobson M. P. and Bence S. J., Mathematical Methods For Physics And Engineering, 2nd ed., Cambridge, 2002 , Section 8.1.3 states the triangle inequality theorem as ||a + b||>=||a||+||b|| and includes a proof. I suggest that the statement of the triangle inequality theorem should be ammended to satisfy the triangle inequality theorem stated by Riley et. al. —Preceding unsigned comment added by 130.56.65.25 (talk) 02:05, 10 December 2007 (UTC)

I agree that the inequality should be instead of . But I can't find which part of the Wikipedia article you are having problems with. Could you please be more specific where in the article collinear vectors and points are excluded? -- Jitse Niesen (talk) 13:47, 10 December 2007 (UTC)

My problem had been with the first paragraph which seems to have been fixed.

The problems that I had yesterday no longer exist. Maybe I was on another planet! My objection was based on the fact that colinear vectors and points are excluded if |a+b| < |a| + |b| but are included with |a+b| =< |a| + |b|. The article now seems quite acceptable to me. Sorry for the confusion. —Preceding unsigned comment added by 130.56.65.25 (talk) 23:47, 10 December 2007 (UTC)