Talk:Triangle inequality

message: Image "Vector triangle inequality.PNG" needs to be edited to correct ||x+y|| < ||x|| + ||y|| to this ||x+y|| <= ||x|| + ||y||.(Jalal0 (talk) 08:27, 5 May 2012 (UTC)
thread_title:  Image correction 

message: 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)
replies: text: 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)
replies: text: Why is it true that? "| |x| - |y| | <= |x + y|"  —Preceding unsigned comment added by 152.23.215.245 (talk) 21:59, 22 November 2008 (UTC)
thread_title:  Consequences unclear and Seem Contradictory 

message: 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)
thread_title:  Intuition 

message: 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)
replies: text: Well-spotted, thanks.  I've made the correction. -- simxp (talk) 11:16, 5 July 2008 (UTC)
replies: text: Yes; good call. This probably has to be corrected on the article on the Cauchy Schwarz Inequality - I just lifted the proof from there...without checking it rigorously myself. Sorry for that...I should have gone through it - I just assumed wikipedia would be correct - force of habit, I guess! The Young Ones (talk) 21:32, 5 July 2008 (UTC)
replies: text: Looking at the Cauch-Swarz article, it must have been corrected seperately since you took the proof.  It doesn't have the 2Re(x,y) line, but the third line correctly uses <= rather than =.  Since the TE isn't the main subject of the C-S article, there's so no need to have the proof longer than necessary in that article, so I'll probably leave it as it is. -- simxp (talk) 13:30, 7 July 2008 (UTC)

text: Yes, I think so, so I changed it. Thanks. -- Jitse Niesen (talk) 22:34, 12 July 2008 (UTC)

text: Every left hand side is compared to the right hand side. Why should the left hand side of the last line be interpreted as being compared with the left hand side of the previous line? That is inconsistent, and likely to be misleading. Waynariffic (talk) 04:23, 17 December 2012 (UTC)
thread_title:  Proof 

message: 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)
replies: text: 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)
thread_title:  Article Deletion 

message: 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)
replies: text: I agree that the inequality should be \|a+b\| \le \|a\| + \|b\| instead of \|a+b\| < \|a\| + \|b\|. 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)

text: 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)
thread_title:  Contrary Definition 

message: I think the illustration of the triangle inequality in the case of equality should be improved such that differently coloured line segments can be identified. Now they have collapsed into one blurry linewhich makes it impossible to see how the quantities are defined. // Jens Persson (193.10.104.171 (talk) 10:47, 2 September 2008 (UTC)
replies: text: I've replaced the figure with one showing three stages of collapse to make clearer what is happening. Brews ohare (talk) 17:10, 1 July 2010 (UTC)
thread_title: Improve the illustration

message: I propose that we add a note saying: The triangle inequality should not be confused with the Parallelogram_law, which is based on the Pythagorean_theorem and specifies precise ratios for the sides of a right-triangle in Euclidean_space.
09:22, 18 November 2008 128.187.80.2
:Such confusion would be objectionable, but I don't understand how it might arise. Brews ohare (talk) 17:12, 1 July 2010 (UTC)
thread_title: Parallelogram Law

message: The following is from the lede:
:That is, this theorem indicates that in Euclidean geometry “the shortest distance between two points is a straight line”, an observation proved for curved lines rather than straight-line segments using the calculus of variations.[3]
This is easily provable without the calculus of variations: if the curve were shorter by ε then one could make a rectification that is shorter by at least ε/2, and then apply the usual triangle inequality repeatedly to obtain a contradiction. 
Moreover, the source provided makes no claim that calculus of variations is important for the proof that the shortest distance is a straight line. The source just says they are presenting such a proof as an example of what can be done with the calculus of variations.
There's no reason to mention the calculus of variations in the lede of this article; it's far from the topic at hand, and there's no reason a reader would need to look it up to understand the triangle inequality. So I am going to remove that mention. — Carl (CBM · talk) 12:27, 2 July 2010 (UTC)
thread_title:  Curves 

message: In Pythagorean theorem there is a section about the converse of that theorem. But what is the converse of the triangle inequality? The article doesn't state nor presents its proof. 187.107.8.106 (talk) 10:19, 2 November 2013 (UTC)
replies: text: I agree. The article should state with proof that if the inequalities hold, then a triangle exists with those sides.208.50.124.65 (talk) 22:10, 8 July 2014 (UTC)
thread_title:  Converse 

message: I am going to finesse the first few sentences to address the following issues:
* The main (first) definition differs significantly from its referenced sources.
* The formula uses x and y without defining them.
* The formula is odd if the reader associates x and y with the nearby diagram of the triangle. Think about the meaning of |x+y|<=|x|+|y| if x and y are the sides of a triangle, and the issue will be clear.
This is a heads-up. Any thoughts? Whikie (talk) 16:13, 25 July 2014 (UTC)
replies: text: It ended up being a small change - just 3 new sentences tying things together and no significant changes to the surrounding text. It is a bit more "mathy" then I would like in the lead, but it isn't too bad. Whikie (talk) 21:11, 25 July 2014 (UTC)
thread_title:  Definition needs a bit of work 

message: thank you for your recent edits.  One quibble, though.  I would like to refer to the measure of an -dimensional facet as a "hyperarea" rather than a "hypervolume".  Specifically, I use "hypervolume" as the general term for the measure of a "face" of a structure (e.g., a simplex), where a face can be of any number of dimensions relative to the number of dimensions of the structure.  However, a "facet" is defined to be a face of one less dimension than the full structure.  When one is referring specifically to this number of dimensions, it is nice to have this more-specific word to use.  Likewise, with "facet" comes "hyperarea".  When referring specifically to this number of dimensions, it is nice to have this more-specific word to use.
Obviously, the more general words could be used instead of "facet" or "hyperarea".  However, math is about precision and I'd like to use the more precise "hyperarea" in this case.  —Quantling (talk | contribs) 13:33, 24 October 2021 (UTC)
replies: text: I understand your motivation, but as far as I can tell this is a word you just made up for convenience, not one that is represented in reliable sources.  In particular, I do not believe there is a meaningful concept to be attached to the phrase "hyperarea of [a thing]".  In contrast, length, area, and volume have fixed meanings, and they are all particular cases of hypervolume (in appropriate dimensions).  As evidence in favor of the theory that this is not a real thing, the word "hyperarea" is used in only two other articles on Wikipedia: Law of sines (it was added in this edit, with no reference supporting its use) and Polyakov action (where it does not appear to be used with the same meaning).  If it were a term with an accepted meaning used by some relevant community, I would expect it to be much better-represented in articles on polyhedra etc. --JBL (talk) 16:57, 24 October 2021 (UTC)
thread_title:  hypervolume and hyperarea