Talk:Metric (mathematics)

message: # in "Definition", it is written "function function" - should we just delete the first "function", or was it intended to replace it with "distance" ?
# I suggest that at least "pseudo" and "ultra" should be defined in the same place than "intrinsic" (much more used, IMHO), i.e. in "Definition"
# "Notes" (I would prefer "Comments") - here the last phrase (2,3,5 => 1) refers to a property "5" which is not defined ; please correct (and, if possible in 1 or 2 lines, write the proof).
# The notion of 'metric induced by a norm' is already used 2 sections before(Examples) so there should be at least a forward reference (maybe better give the definition in a subsection of "examples")
— MFH: Talk 12:59, 19 Apr 2005 (UTC)
thread_title:  bug fixes and cleanup suggested 

message: The articles Metric (mathematics), Distance, Metric space, and maybe something else, have evolved independently after mathMarti decided to split them. May be his intentions were good, but now the articles create a confusing mess rather than a coherent mathematical discourse. Please someone can initiate a microproject to put some order/system into this topic? `'Míkka 17:48, 8 October 2007 (UTC)
replies: text: See my simultaneous contribution to the section on the split. PJTraill (talk) 19:32, 11 March 2012 (UTC)
thread_title:  Content syncronizing 

message: I have made several changes in metric and metric space. Most of them should speak for themselves, but perhaps I should explain why I have removed all references to hemimetric and prametric. Most people are reluctant to remove something just because they have never heard of it. So am I, but this was a special case.
* The only reference given for prametric is Arkhangelskii and Pontryagin: General Topology I. My library doesn't have the book, and it's not on Google books either, therefore I can't check that the strange word "prametric" really appears in the book. Google has 1660 hits for it. Each of the first 100 hits is either an obvious misspelling for "parametric", or it's on Wikipedia or a clone. Google Scholar has 94 hits, every single one is an obvious misspelling for "parametric". Google books has, surprise, one relevant hit among 25 misspellings: Ruben Aldrovandi and J. G. Pereira, An Introduction to Geometrical Physics. So that's a total of two books defining the term. (The book on Google Books doesn't seem to do anything with it except that it derives a topology.)
* The only reference given for hemimetric is PlanetMath. PlanetMath does not give a reference at all. The presyllable "hemi" is usually used as a synonym for "semi". The concept itself is often referred to as pseudo-quasi-metric, which is a bit longer but logical once pseudometric and quasimetric have been defined. Google has 852 hits. Some of them are for "m-hemimetric", whatever that is, and I suspect that many are variants for "semimetric". Google Scholar has 0 hits. Google Books has 2 hits: One for "m-hemi-metric", and one for "quasi-hemi-metric", a combination that makes no sense with our definition.
The only frequently used terms for generalized metrics appear to be:
* pseudometric
* semimetric — though very often in the sense of pseudometric
* quasimetric.
Since all combinations of the axioms can be expressed by combinations of pseudo-, semi- and quasi-, there is no reason why Wikipedia should coin a strange new term such as "prametric". (Or help the translator of a Russian book to do so.)
To be absolutely sure, I made a number of searches in publications. Numbers are for search without hypen/with hyphen, e.g. prametric/pra-metric.
word        	Math. Reviews	Zentralblatt
prametric   	0/0          	0/0
prometric   	0/0          	1/0
premetric   	21/14        	12/15
hemimetric  	2/3          	1/0
pseudometric	821/311      	386/254
semimetric  	236/112      	98/84
quasimetric 	138/273      	71/253
I think these results show clearly that it is seriously misleading to direct unsuspecting readers to the articles on prametric and hemimetric. The remaining question is whether to delete them altogether. I think there should be something like a notability criterion for definitions, and I suspect they wouldn't pass.
Perhaps some people who actually work in topology and functional analysis, or in related disciplines, can check whether the passages about the two definitions of semimetric really describe usage in their respective fields. I have also asked the MSC2010 group to clarify what the MSC2000 category "54E25 Semimetric spaces" is about. --Hans Adler (talk) 02:56, 22 November 2007 (UTC)
thread_title:  Generalized metrics 

message: User:Valandil211 started changing html to inline TeX in the article.  I strongly believe that there is no consensus for it, and I'd like to get some support to tell him to stop doing this. --Bdmy (talk) 12:48, 20 April 2009 (UTC)
replies: text: I agree that there is no consensus, and I don't agree with the change. --Hans Adler (talk) 14:38, 20 April 2009 (UTC)

text: I also agree that there is no consensus, and I strongly prefer the inline TeX. DVdm (talk) 20:36, 20 April 2009 (UTC)
thread_title: Inline TeX

message: "In differential geometry, the word "metric" is also used to refer to a structure defined only on a differentiable manifold which is more properly termed a metric tensor"
This claims that a differential manifold is more properly termed a metric tensor.  I suspect "which is" was intended to refer back to the structure.
Perhaps: "...used to refer to a structure defined only on a differentiable manifold. Such a structure is more properly termed a metric tensor"
Gwideman (talk) 21:07, 4 August 2012 (UTC)
replies: text: I see two problems here: a missing comma before "which", and ambiguity as to which noun phrase it qualifies. (It is intended to refer back to the "structure" mentioned.) Definitely needs rephrasing to avoid the ambiguity. I have done so. — Quondum☏ 23:03, 4 August 2012 (UTC)
thread_title:  Clarification needed in intro 

message: The word "induce" is used on this and several related pages.  Does this have some specific technical meaning?  Is it related to the word "induction" in logic?  Or is it perhaps a synonym for "implies", or just "results in" or ????   Gwideman (talk) 21:48, 4 August 2012 (UTC)
replies: text: Not logical induction. In a mathematical context it seems to mean "implies the existence of". Or more informally, "produces". Though I don't see this precise meaning in the online dictionaries that I browsed. — Quondum☏ 23:13, 4 August 2012 (UTC)
thread_title:  "induce" -- what does that mean? 

message: I think this new material is not appropriate, at least in the present form:
* Misnomer: The condition that is relaxed in comparison to quasimetrics is positive definiteness, not positivity. Positivity, at least as used in this article, just says that d(x,y) >= 0 for all x,y. This actually holds.
* Apparently d(x,x) = 0 is also required, although it's not stated explicitly. Thus the notion is identical with that of pseudoquasimetric or hemimetric, already discussed further down in its own section.
* The example is excessive compared to the other equally or more important generalisations. 
I guess the maximum that we need here is a mention that someone has called pseudoquasimetrics  non-positive quasimetrics and maybe a link to the book for an example of such a thing. Hans Adler 20:50, 12 February 2013 (UTC)
thread_title:  New section Non-positive quasimetrics 

message: As it currently reads in Wikipedia, the distance induced by a quasi-norm is a semi-metric, and a distance induced by a semi-norm is a quasi-metric. This is confusing, and something should be done to fix this. Whatever the solution is, it should be consistent between generalized norms and generalized metrics.
To show that mathematicians have used the term quasi-metric to refer either to a lack of symmetry, or to a relaxed triangle inequality, a brief search provides the following papers
Quasi-metrics as relaxed triangle inequality:
* Quasi-metric and metric spaces (2006)
* A new approach to function spaces on quasi-metric spaces (2004)
* Fixed point theorems in generalizing spaces of quasi-metric family and applications (2001)
Quasi-metrics as lacking symmetry:
* Fixed Point Results in Dislocated Quasi Metric Spaces (2014)
* Quasi-metric properties of complexity spaces (1999)
* Quasi-Metric Spaces, Quasi-Metric Hyperspaces and Uniform Local Compactness (1999)
* Weighted quasi-metrics (1994)
* On completeness in quasi-metric spaces (1988)
* Cauchy-sequences in quasi-pseudo-metric spaces (1982)
* A note on quasi-metric spaces (1976)
* On quasi-metric spaces (1931)
--Kaba3 (talk) 21:14, 1 December 2014 (UTC)
thread_title:  Terminology is inconsistent with semi-norms and quasi-norms 

message: Ether the definition might be wrong. Based on https://en.wikipedia.org/wiki/Ultrametric_space , first paragraph , if y is between z and x on the real number line, then d(x,y)≤max(d(x,z),d(z,y)) seems like the correct version. However due to my naivety with Maths, I have not made any changes to ether page concerning this. Regardless, I believe that one of the two pages is wrong concerning this.  — Preceding unsigned comment added by Ruin052 (talk • contribs) 02:30, 11 December 2014 (UTC)
replies: text: The real number line is a metric space, but not an ultrametric space. —David Eppstein (talk) 02:33, 11 December 2014 (UTC)
thread_title:  Definition of Ultrametric 

message: The lead paragraph suggests that metric and metric tensor are the same.  We have two separate articles for these topics.  Is the lead correct?  If that's the case, then these articles should be merged.  Is metric tensor applied more typically in the context of differentiable manifolds, whereas metric is the more general concept, used usually in discrete spaces?  In other words, is there a distinction here, which should be pointed out?  If not, why not merge?  — Preceding unsigned comment added by 70.247.173.205 (talk) 18:15, 28 February 2016 (UTC)
replies: text: That is a misreading. They are not the same, and should not be merged. In particular, there are many examples of metrics and metric spaces that do not come from a metric tensor. —David Eppstein (talk) 20:01, 28 February 2016 (UTC)
thread_title:  Metric and metric tensor 

message: How does the Minkowski metric fit in? It is called a metric on Minkowski space, but it does not appear to satisfy any of the definitions in this article. Each of the special flavors (pseudometrics, quasimetrics, metametrics, semimetrics, premetrics) include semi-positive definiteness (d(x,y)\ge 0). But the Minkowski metric is not semi-positive definite. Klaas van Aarsen (talk) 12:30, 26 October 2019 (UTC)
thread_title:  Minkowski Metric 

message: The picture right on the top of the page has this description: “An illustration comparing the taxicab metric to the Euclidean metric on the plane: According to the taxicab metric the red, yellow, and blue paths have the same length (12). According to the Euclidean metric, the green path has length 6 \sqrt{2} \approx 8.49, and is the unique shortest path.”
This is misleading, because the red, yellow and blue paths have the same length in taxicab and Euclidean. The important thing is that with taxicab metric, all the four lines, including and especially the green one, have length 12 and are the shortest path between the two points. The current description doesn't seem to mention anything about the taxicab metric that is actually important. However I'm hesitant to edit it since it's been there for so long without anybody noticing. Is there an important reason why it's like this?  Michal Grňo (talk) 09:26, 14 January 2020 (UTC)
thread_title:  The taxicab×Euclidean figure has a misleading caption 

message: The comment you supplied when reverting my edit is confusing to me, which leads me to the belief that my edit to the article was confusing to the reader.  I will try for more clarity with my next attempt.  Thank you for looking out for the quality of Wikipedia.  —Quantling (talk | contribs) 12:40, 11 August 2021 (UTC)
replies: text: You need (1) to make clear the relevance of what you are talking about to distance functions, rather than to Euclidean spaces (a different thing, and not the topic of this article), and (2) to back up all claims in whatever you add by reliable published sources. Also, obviously (3) to say something that is meaningful and correct mathematically. None of 1,2,3 was evident in what you added. —David Eppstein (talk) 16:30, 11 August 2021 (UTC)
thread_title:  Euclidean metric as the motivation for metrics generally