Talk:Metric (mathematics)

Range implied by first condition in definition
The first sentence in the definition of the metric explains about the range being positive, immediately explaining that 'distance' can't be negative. Then, property 1 of a metric says it's positive, which is actually specifying its range again. And then it says that non-negativity follows from the others. Sounds a bit superfluous to me. Mar10dejong (talk) 08:15, 8 March 2016 (UTC)


 * See below. my private opinion: non negativity is too fundamental to not mention it (and later the other axioms are removed), ps next time add new sections yat the end of the page  :)  WillemienH (talk) 11:19, 8 March 2016 (UTC)


 * I think both of you are emphasizing different important ideas. As presently stated, it is NOT mathematically elegant (and actually inaccurate) to claim that all three axioms are responsible for the result that distinct points have positive distances.  Here is why: Because the function is defined to be non-negative, d(x,y) can only be zero or positive.  Now, axiom 1 rules-out distinct points being 0, thus forcing their distance to be positive--without axioms 2 & 3.


 * That said, what it SEEMS that one of you is trying to say is that if d were defined as a real-valued function without the non-negative range restriction, then all three axioms together would imply the positive criteria. I'm going to make a change to this effect to see if we can't make both distinctions clear.


 * By the way, as one side-note, we would call this a non-negative real-valued function, not a positive definite function. The term positive definite is a linear algebra term for matrices; providing that distinction and link sends the reader off on a very unclearly connected idea.  Let's just keep this as saying a non-negative function. Dr. Belnap 15:49, 7 May 2021 (UTC)

Split metric space into metric (mathematics) and metric space
I split metric space into metric (mathematics) and metric space. Reasons were MathMartin 12:27, 8 Apr 2005 (UTC)
 * metric space article has become too long
 * norm (mathematics) and normed vector space are separate articles too
 * different metrics can be used to define the same metric space (depending of course on how one defines sameness)


 * I think this splitting up is a good idea, but more cross-links are needed between these two pages (and they should refer to precise sections and/or examples, so section titles should be well-thought).
 * I would only criticize a bit your last point: In mathematics, "the same" should mean equal, IMHO. "Equivalent metrics" are now defined, they define the same topology, i.e. the same topological space, but not the same metric space (defined as a certain 2-tuple).

&mdash; MFH: Talk 13:28, 19 Apr 2005 (UTC)


 * Section linking between articles is a very bad idea, because the links are prone to breaking. If section linking cannot be avoided the two articles should be merged again. What do you mean by same should mean equal ? MathMartin 13:44, 19 Apr 2005 (UTC)
 * Section links need not get broken if you add anchors to the sections, and if people realise the implications of the anchors :( PJTraill (talk) 19:32, 11 March 2012 (UTC)


 * I think that splitting was perhaps unfortunate, or could have been done better, as witness the section asking for synchronisation of content. I would suggest that if articles are spilt there should be:
 * No more overlap between them than is necessary to make the individual articles readable without excessive jumping back and forth.
 * An easily formulated principle enabling an editor to decide which content belongs where, perhaps in an HTML comment.
 * A clear indication at the beginning of each article which other article(s) there are, and of what content can be found where, consistent with the principle guiding the split.
 * Also (redundantly) a reference under See also to the related articles.
 * In the case of this particular split I suppose the principle could be that variants on the concept and properties of the function go in Metric (mathematics) while properties of spaces go in Metric space. I am afraid I don’t have time at the moment to clean things up on this basis &hellip; perhaps later, perhaps someone else? PJTraill (talk) 19:32, 11 March 2012 (UTC)

bug fixes and cleanup suggested
&mdash; MFH: Talk 12:59, 19 Apr 2005 (UTC)
 * 1) in "Definition", it is written "function function" - should we just delete the first "function", or was it intended to replace it with "distance" ?
 * 2) 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"
 * 3) "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).
 * 4) 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")

I recently split this article from metric space so most of the bugs are probably from me. I had intended to do a cleanup in the next few days. All your suggestions seem reasonable so go ahead. However I would keep the old section title Notes as it is more common in math articles on wikipedia.

As a sidenote, I think it is uneccessary to discuss such small modifications on the talk page. The next time you should be more bold and just do the changes. MathMartin 13:27, 19 Apr 2005 (UTC)

remove fraction
I changed


 * da( x, y ) = &Sigma; ai pi(x-y) / (1 + pi(x,y))

to
 * $$d(x,y)=\sum_{n=1}^\infty \frac{1}{2^n} \frac{p_n(x-y)}{1+p_n(x-y)}$$

so I changed that ai coefficent to a specific geometric series. That's what's in my book, and I guess it's needed to ensure convergence of this sum -Lethe | Talk 05:49, July 11, 2005 (UTC)

Content syncronizing
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)
 * See my simultaneous contribution to the section on the split. PJTraill (talk) 19:32, 11 March 2012 (UTC)

Generalized metrics
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: 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.)
 * pseudometric
 * semimetric — though very often in the sense of pseudometric
 * quasimetric.

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. 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)

PS: I was drawn into this because preclosure operator was misspelled as praclosure operator and had a link to prametric space. "Praclosure" has 0 hits in Google Scholar and Google Books, but 167 on Google. All on Wikipedia, including clones and translations. --Hans Adler (talk) 03:01, 22 November 2007 (UTC)

Axioms of a metric
Correct me if I'm wrong but I believe that the first condition in the definition is redundant as it follows from identity of indiscernibles (condition 2) and triangle inequality (condition 4) Jergosh (talk) 20:36, 13 May 2008 (UTC)

A little update: I checked the definition with a book I'm using at my university (not a definitive resource as it is printed locally) and it states that there are three 'axioms of a metric' (identical to axioms 2, 3 and 4 here on wikipedia) and that nonnegativity follows from them (apparently all three). I will try to research this further but I have limited access to sources in English. Jergosh (talk) 20:52, 13 May 2008 (UTC)


 * It's easy to prove 1 from 2-4 (and an easy counterexample with two points shows just 2 and 4 are not enough). But this doesn't mean we need to drop 1. --Hans Adler (talk) 23:38, 13 May 2008 (UTC)


 * Why keep it if it's redundant? Perhaps it would be better style to keep 2-4 as axiom and remark that nonnegativity follows from them (i. e. is not an axiom but a useful property). Jergosh (talk) 06:41, 14 May 2008 (UTC)


 * It becomes important when you consider certain subsets of the axioms; for many such subsets there are names. Positive definiteness is even specifically mentioned in the article. Also in some axiomatisations non-negativity is built in even before the first axiom is mentioned, by saying that "a metric is a function d: X &times; X → R0+ satisfying..."
 * I am not fundamentally opposed, I just don't think it would be a clear improvement to drop the axiom. --Hans Adler (talk) 11:51, 14 May 2008 (UTC)


 * Consider the following example: Let $$X$$ be the space of possible states of a certain machine (for example, $$X=\{0,\dots,n-1\}$$ if the machine is an elevator in a building with $$n$$ floors). Let $$d(x,y)$$ be the infimum energy needed to take the machine from state $$x$$ to state $$y$$. This function satisfies the triangle inequality but is non-symmetric: rising the elevator always costs a lot of energy, but lowering it should cost less, or zero, or maybe we can even obtain energy from that operation. The important thing is that we can't obtain energy from travelling a closed cycle. So the relevant positivity axiom is $$d(x,x)\geq 0$$ (which of course implies that $$d(x,x)=0$$, since it is clear that $$d(x,x)\leq 0$$ because we can go from $$x$$ to $$x$$ at zero cost by doing nothing). The most similar concept described in this article is that of quasimetric, but it does require $$d(x,y)\geq 0$$. So I wonder if, in the applications of that concept, one can do with only $$d(x,x)=0$$. By the way, I find the term quite descriptive, think of it: quASIMETRIC :-P. Marcosaedro (talk) 19:07, 28 January 2013 (UTC) (Remark: Here when I say "energy" I mean 'useful energy', so heat doesn't count. So even if energy is always conserved, useful energy is conserved or diminished.)
 * I added a reference for only requiring $$d(x,x)\geq 0$$. Smyth, M. (1987). "Quasi uniformities: reconciling domains with metric spaces". And I merged the example explained above into the text of the article. Marcosaedro (talk) 19:39, 12 February 2013 (UTC)

Does anybody know how a Minkowsky space would fit in here? It seems that it drops 1,2, and 4 but I never found a mathematical name. —Preceding unsigned comment added by 76.193.219.196 (talk) 22:11, 5 October 2009 (UTC)


 * Notice that if you can go from an event $$e_0=(t_0,x_0)$$ to $$e_1=(t_1,x_1)$$ (i.e, if $$t_1-t_0\geq\sqrt{(x_1-x_0)^2}$$), then the interval $$s(e_0,e_1)$$ is the maximum of the durations of all possible trips from $$e_0=(t_0,x_0)$$ to $$e_1=(t_1,x_1)$$. This maximum is achieved iff the trip is the straight trip, and can be calculated as $$\sqrt{(t_1-t_0)^2-(x_1-x_0)^2}$$. Separation satisfies a reversed triangle inequality $$s(x,z)\geq s(x,y)+s(y,z)$$. And it is non-symmetric, in fact, whenever $$s(x,y)$$ and $$s(y,x)$$ are defined, it follows that $$x=y$$.Marcosaedro (talk) 19:42, 28 January 2013 (UTC)

Inline TeX
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) --Bdmy (talk) 17:36, 20 April 2009 (UTC)
 * I agree that there is no consensus, and I don't agree with the change. --Hans Adler (talk) 14:38, 20 April 2009 (UTC)
 * I also agree that there is no consensus, and I strongly prefer the inline TeX. DVdm (talk) 20:36, 20 April 2009 (UTC)

Clarification needed in intro
"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)


 * 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)

"induce" -- what does that mean?
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)


 * 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)

New section Non-positive quasimetrics
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)

Terminology is inconsistent with semi-norms and quasi-norms
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)

Definition of Ultrametric
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)
 * The real number line is a metric space, but not an ultrametric space. —David Eppstein (talk) 02:33, 11 December 2014 (UTC)

Metric and metric tensor
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)
 * 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)

Quasimetric inducing the Sorgenfrey Line
The article mentions offhand, in the section on quasimetrics, that the quasimetric inducing the Sorgenfrey Line may defined as
 * d(x, y) = 1 if x < y

but that this 1 may be replaced with infinity or with $$1+10^{(y-x)}$$. These specific three numbers, 1, infinity, and $$1+10^{(y-x)}$$ seem sort of arbitrary and bewildering; does anyone have a source for them?

Minkowski Metric
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)

The taxicab×Euclidean figure has a misleading caption
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)

Euclidean metric as the motivation for metrics generally
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. — Q uantling (talk &#124; contribs) 12:40, 11 August 2021 (UTC)
 * 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)

The added value (#1 and #3) was supposed to be an easier to read version of: a function $d: X×X → R$ is a metric (aka a distance function) if and only if for every subset $S ⊆ X$ with $|S| ≤ 3$ there exists an injective function $m: S → R^{|S|−1}$ such that for all $a, b ∈ S$ it is the case that $d(a, b) = d_{E}(m(a), m(b))$, where $d_{E}$ is the usual Euclidean distance function. Maybe I can say that "d is a metric whenever, for any subset of three points or less, it is the case there is a subset of Euclidean space of the same size that yields the same pairwise distances." (Your input here would be much appreciated.) And as to #2, you are absolutely right; I will look through my textbooks for the source. (Your input here would also be much appreciated.) — Q uantling (talk &#124; contribs) 19:49, 11 August 2021 (UTC)